Orbital Dynamics in 3 He-A *l

Transcription

1 772 Progress of Theoretical Physics, Vol. 63, No. 3, March 1980 Orbital Dynamics in 3 He-A *l --I-Iomogencous Case-- Haruo TAKAGI and Kazumasa MIYAKE Department of Physics, Nagoya University, Nagoya 46 4 (Received November 15, 1979) The orbital dynamics of the homogeneous superfluid 'He-A is investigated through microscopic calculation of the response function for the l-vector of the Cooper pairs. It is shown that Leggett and Takagi's phenomenological theory is justified in the region J (T)r:;}> 1 and wr<;:l. However a critical exponent of the relaxation time of the l-vector in the region.d (T)r<;:1 is zero in contrast with their theory. The intrinsic angular momentum has the same form in the two frequency regions. I. Introduction The orbital dynamics, as well as the spin dynamics, is one of the most important problems in superfluid 3 He-A. It is related to the problem about the existence of a macroscopic orbital angular momentum. In homogeneous case the orbital dynamics has been studied by many authors; Cross and Andersonll (CA), Pethick and Smith, 2 J Leggett and Takagi 3 J (LT), \Volfle, 4 l and so on. It is shown in their works that in the hydrodynamical region the dynamical behavior of the Z-vector is overdamped while in the collisionless region it is oscillatory. The motion of the Z-vector is different from that of the d-vector in the spin dynamics which is oscillatory in both regions. In the hydrodynamical region, the equation of motion of the Z-vector which represents the orbital configuration of a Cooper pair can be written in the form 3 l dl dl Lo-=gDl X z-!j.l X, dt dt (1) /). is an orbital viscosity, L 0l an intrinsic angular momentum and gdl Xz a dipole torque. It depends on the value of L 0 / /). whether the motion is oscillatory or relaxational. An experiment by Paulson, Krusius and Wheatley 5 J observed an exponential relaxation, so that it can be concluded that /). is much larger than L 0 On the basis of an intuitive picture analogous to the spin dynamics, L T obtained r is a lifetime of the normal-state quasiparticle at the Fermi surface and (2) *l A preliminary report was published in Prog. Theor. Phys. 60 (1978), 1924.

2 Orbital Dynamics in 3 He-A 773 r x is a phenomenological relaxation time of order of 1:". In the region, L1 (T) r )> 1, smce gnr is much larger than Xorb/rx, /}.. equals gnr in agreement with the result of CA. However in the gapless region, L1 (T) r~1,!j. is Xorb/rx and proportional to (1- T /Tc) 112 in contrast with others. Recently, Nagai 6 l justified L T's theory in the region L1 (T) r)>1 and wr~1 within the framework of matrix kinetic equation. According to the microscopic calculations by Crossn in the collisionless region, the intrinsic angular momentum is given by (3) which is proved to hold also in the hydrodynamical region by Nagai. 6 l Since the value of L 0 is much smaller than that of /}.., the motion of l is a relaxational one in contrast with an oscillatory one of d. The difference between the motion of l and that of d, as pointed out by L T, comes from the fact that, for orbital dynamics, there is no local conserved variable such as spin density. We discuss this problem in the last section. In this paper, we investigate the homogeneous orbital dynamics in the A-phase both in the hydrodynamical and in the collisionless region. For this purpose, we calculate the response function for the 1-vector microscopically over the two regions. We assume that the d does not vary and consider a model Hamiltonian which includes the dipole interaction as well as the kinetic energy and the BCS like pairing interaction. In the collisionless region our result is equivalent to that obtained from the kinetic equation approach.sl. 4 l However in the hydrodynamical region it differs a little from others; in the gapless region (Ll (T) r~1), a critical exponent of the relaxation time of the l is zero which is different from both LT and CA. It is shown that the intrinsic angular momentum L 0 has the same form both in the hydrodynamical and in the collisionless region. 2. Formulation For superfluid 3 He, d (n) -vector is introduced as follows: (4) d (n) Is normalized as (5) In the A-phase, the form of d (n) is given by (6)

3 774 H. Takagi and K. Miyake d; is a unit vector in the spin space, and a 1 and ct 2 are real orthogonal unit vectors in the coordinate space. Then the Z-vector, which describes the orbital configuration of the Cooper pair, is given by In equilibrium, l is forced to lie along d by the dipole energy. We consider the case that d and l are parallel to the y-axis and a 1 to the z-axis and a 2 to the x-ax1s. From Eqs. (4) r-j (7) l can be written in the form l = id X D* I (D. D*) ' (8) (7) (9) As we consider a small deviation of l from equilibrium state, the relevant components of l are l, and Zx. Then we define their operator forms by omitting the averaging procedure in the relation (8) as (10) we have performed linearization and used the fact that D 1n equilibrium is given by (11) The form (11) can be obtained from Eq. (9) with a gap equation (12) which equals../3j2i1(k,+ikx) in equilibrium. Dipole interaction between nuclear spins is incorporated into Hamiltonian, by taking -1/2gn (l d) 2 as the effective interaction as H c 1 z' 2 z' 2) n=--gdly =--gd - z- X ' 2 2 (13) we have used the fact that l is a unit vector and the assumption that d is fixed along the y-axis. Substituting the operator form (10) into (13), we obtain the total Hamiltonian including the dipole interaction as well as the kinetic energy and the BCS like pairing interaction:

4 Orbital Dynamics in 3 He-A 775 (14) It should be remarked that the dipole term does not affect the gap equation (12). 3. Response functions Making use of the operator form of l (10) and the Hamiltonian (14), we calculate the response function for the Z-vector: wheer a, f]=z, x. In order to obtain the response function, we calculate by means of thermal Green's function. Dominant contributions to XaP come from the perturbation series corresponding to the Feynman diagrams in Figs. 1 (a) and (b). The terms represented in Figs. 1 (c)~ (e) vanish after momentum summation. The other terms can be neglected due to a small factor of (w,/cf). That is, the term represented in Fig. 1 (f) is of order w,/ 2F compared to the term represented in Fig. 1 (g) because the interaction is restricted within the thin region around Fermi scuface. Then we obtain X ail as follows: <=> + <=:><:::=> +... (a) <=> + <=:><:::=:> +... (b) Fig. 2. (c) (d) (c) (f I (g) Fig. 1. Fig. 1. The series (a) and (b), are the diagrams contributing to X1 and X2 respectively. The terms involving the diagrams (c)~ (e), vanish after momentum summation. Fig. 2. The Feynman diagram which is taken into account for the self-energy.

5 776 H. Takagi and K. 1\lliyalw Xzz (Wn) = Xxx (Wn) = X1 (Wn) + X2 (Wn), Xzx (wn) = - Xxz (Wn) = i {Xl (cun) - X2 (wn)}, X (w ) = Vo 2!1 (Wn) 1 " 6 1' 1-Vo(1-Vogn/3L12)fl(w,.) x,(w) = Va' ~ f,(wn).... " V 0 (1-V 0g n/3l1') f, (cu,.) (16a) (16b) (17a) (17b) / 1 (w,.) -T 2.:: 2.:: py'g (p, w,) G (-p, Wn ~w,), w, p (18a) J, (wn) =T 2.:: 2.:: Pv 2 G (p, U),) G (-p, ~wn -wv). "'' p (18b) We can take account of the effect of collisions between quasi particles through the self-energy of th~ type in Fig. 2. After analytical continuation of iwn from the upper half of the complex plane, };, 2 (w) can be written as follows: -GR (~, p, x+ w)gr ( -~, p, x)] +[tanh x-w -tanh~ ] 2T 2T X GR (~, p, x) GA(- ~' p, x- w)}, (19a) f, (w) = sdsj_ S"' d~ soo c{x_n (~) Pv 2 {tanh X [GA(- ~' p, x) GA(~, p, x- w) 4n -w. -oo 4nz 2T x GR (- ~' p, x) GA (~, p, x ~ w)}, (19b) and N (~) is the density of state and r is a collision time between quasiparticles. It is easily seen from Eqs. (16) ~ (19) that within the approximation of particle-hole symmetry X 1 (w) equals X 2 (w). Therefore Xzx and Xxz vanish, which means that l-vector does not precess at all. Then we expand the density of state as (20)

6 Orbital Dynamics in 3 He-A 777 First of all we consider the problem near the transition temperature, that is; T~A (T), 1/r, and in the frequency region w«;_t which is not so strong a restriction for practical interest near T,. Substituting Eq. (19) rv (21) into Eq. (17), X1. 2 is reduced to (see Appendix A) (21) (22a) (22b) (23) and La= N (O)J~ J'"' d~l_tanh_f_ 8eF -w, ~ 2T =~A~ln 2rwc 16EF 2 nt Introduction of particle-hole assymmetry makes L 0 finite which causes the l to precess around the z-axis. So we call L 0 an intrinsic angular momentum. From derivation of Eq. (24), it is seen that for the whole temperature region La is given by 3 zl:;~z~ E La=-J k -tanh-- 4 '" v E 3 2T ' E= JP:+ ILl~V. Since Eq. (22) holds both in the hydrodynamical and in the collisionless region, L 0 has the same form over both regions. The expression of L 0 (25) agrees with the result by Cross 7 > and Nagai. 6 J Since La can be practically neglected due to a small factor (A/eF) 2, we ignore it in this section. Let us consider the structure of x=x 1 = x 2 in the following case. (1) In the region A~l1/r-iwl, from Eqs. (22) and (23), we have (24) (25) (w) _ 1/2 X - 9n(-iw)/(-iw+1/r) +3/4 Xarb(-iw) (-iw+1/r) +gd ' (26) (27)

7 778 H. Takagi and K. }.;Jiyake and = (~) 1 ; 2 rr~n (0).d Xorb 2 32T (28) The denominator of Eq. (26) is equivalent to Eq. (7 14) of L r) except the factor 3/4 in front of Xorb if 'f"k is regarded as -r. It agrees with Nagai's 6 l result. Since in the collisionless region X (rjj) has a pole oj 0 of order of.d as we shall see later, we consider Eq. (26) in the hydrodynamical region. Then the form of X (cv) is reduced to (29) which has a pole (!)0 as 9n Wo= g n '( + 3/4 Xorb 'f"- 1 (30) (2) In the region 1~11/-r-i(J)I (gapless region), Eq. (22) 1s reduced to 1/2 x (w) = iwrrn (0) A 2 j8t + g n (31) Then we obtain a pole W 0 as (32) which lies in the hydrodynamical region. From the expressions of (29) and (31), the orbital viscosity 11 1s given by _ {gn-r+3/4 XarbC 1 f!.- rrn(o).d 2 /8T for.dr-::?>1, for.d-r ~ 1. (33a) (33b) Outside the gapless regwn.d (T) r-)>1 or 1-T /Tc:::f>10-4, gn-r is much larger than Xorb'f"- 1 Hence the relaxation time of lis gn-rfgn and proportional to (1- T /Tc) 112 in agreement with the result of CA 1 l and LT. 3 J However in the gapless region Ll(T)r-~1, the relaxation time is rrn(o)a 2 /8Tgn. Since near Tc both 9n and A 2 are proportional to (1- T ftc), the critical exponent of the relaxation time is zero. Next we consider the structure of X (w) in the collisionless region. From Eqs. (16) ~ (18), we obtain X (w) over the whole temperature region as (see Appendix B) X (w) = _l {~.d 2N (0) L: l_tanh~(l_- ~p 2 + p 2 2 1Apl~~) + g } -~. 2 2 p E 2T 2 2 y y 4E2 - (!) 2 D (34)

8 Orbital Dynamics in 3 1-Ie-A 779 It is shown in Appendix B that the denominator of 'X is the same as lhs of Eq. (6 28) in LT 31 except a dipole term. In the low temperature limit, the expression of (34) is reduced to (35) In the denominator of (35), the dipole term can be neglected compared with the second term except the region T /T c<l0-3 Therefore 'X has a pole as ( 2 )1;2 [ 2r.::1 J-112 OJ 0 = 3 nt ln nt- (36) 111 agreement vvith the result of \Volfle.' 1 Near the transition temperature, we obtain 'X from Eq. (34) or Eq. (22) as (37) Since the pole of (35) is of order of.::1, gd can be neglected except the region (1- T /T c) <10-8. Thus 'X is reduced to l+.q]}-l +-(1-.Q ) (1-5.Q ) ln --, 2 1-.Q (38) It is easy to see that the denominator of (38) is not zero for any real oj. vve find through numerical calculation that the imaginary part of the response function (38) has a peak at U) = 4/5 /3/2.::1 with a width of order of.::1. Therefore near the transition temperature a flapping mode is not so well defined as in the low temperature limit. From Eq. (31) we put no restriction on cu but uj<!(t, it is concluded that a flapping mode does not exist in the gapless region. region. 4. Equation of motion for the Z-vector In this section we derive the equation of motion for l in the hydrodynamical Let us start with the Mori equation 81 (39)

9 780 H. Takagi and K. Miyake iwa/3= Cla, l,) (l, l);i, (fja/3 (s) =(fa (s), f/) (l, l) ;i and the random force J" is defined by The scalar product (l "' l ~) is usually defined by In homogeneous case the only variable to be considered 1s l because of the assumption that d does not vary. Therefore the correlation time of random forces must be much shorter than the characteristic time of l. Hence Eq. (39) becomes Markoffian, (40) In order to derive the coefficients of l, we make use of the identity:. l =!_(x(z) -x(o)x(o)- 1, z-zo)+ip(z) z (41) and We have obtained the response function Xa/J(co) in 3 as follows: and = ~ t_ itlw=- Law +9~ + - itlw +~ow+ g J (42a) (42b) From Eqs. (40) rv (42), the equation of motion for l, and lx can be written as

10 Orbital Dynamics zn 3!-le-A 781 (43) L 0 and p are given by - {gnr+3/4xorb C 1 /1- nn (0) 1Pj8T for for (see Eqs. (23) and (24)). 5. Concluding remarks We have calculated the response function of l both in the hydrodynamical and m the collisionless region. The expression for "b Eq. (22), holds over the two regions. Our results are almost in agreement with others and LT's 31 theory has been justified in the region Llr)>1 and wr<t1. We have shown that the intrinsic angular momentum L 0 has the same form over the two regions. However in the gapless region our result is different from others; that is, the critical exponent of relaxation time of l is zero. The approximation used in our calculation is rather simple in the hydrodynamical region. We have taken account of the effect of collisions between quasiparticles only through the imaginary part of the self-energy for the Green function. A vertex correction need not to be considered due to a small factor of ((!Jc/ cf). If we calculate the dynamical spin susceptibility within the same approximation, \Ve obtain a pure imaginary pole in the hydrodynamical region. As a result the motion of d-vector becomes relaxation not oscillation which is derived theoretically9l.iol and observed experimentally.w In calculation of the dynamical spin susceptibility we must consider a vertex correction.12l The difference between the orbital dynamics and the spin dynamics reflects the fact that in orbital dynamics there 1s no local conserved variable coupled to the Z-vector such as spin density. Acknowledgements In the course of this work we have benefited from conversations with Professor T. Usui, Professor M. Ishikawa and Professor Y. Kuroda. Appendix A --Derivation of Eq. (22) from Eq. (17) - We rewrite f 1 (w) in Eq. (19a) as follows:

11 782 H. Takagi and K. 1\fiyake -GR (~, p, x)gr ( -~, p, x-oj)] +(tanhx'_--:_o)_tanh x )[GR(~,p,x)GA(-~,p,x-oJ) 2T 2T -GR (~, p, x)gr ( -~, p, x-w)j } (A 1) Substituting the Green function (20) into (A 1), the first term is reduced to Near the transition temperature (T)>Ll, 1/r) and in the frequency region u)<{.._t, (A 2) becomes The gap equation,can be written in the form, Using the expression for F analogous to G in Eq. (20), the gap equation is reel ucecl to (A 4) Subtracting (A 4) from (A 3), we have 1 N (0) w sw, ( 1 ~ 1 _z ~ ) 1 1= d~ -tanh - -cosh ---, Va 24eF -w, ~ 2T 2T 2T (A 5) we have used the form for the density of state in Eq. (21). The second term of Eq. (A 1) is given, up to first order of wjt, by

12 Orbital Dynamics in 3 IIe-A 783 which is reduced to 12=- N(O)_c~ S clj2_ sd~py2 ~~~P_I~=_(w+i/r) T 4rc 8(co+i/r) [~ 2 -!-IL1pl 2 - (uj-!-i/r) 2 /4] +}V_(O)_uJ_ S"'' d~ cosh_ 2 _f_. 48TeF -w, 2T (A 6) From Eqs. (A 5) and (A 6),! 1 (co) becomes (A 7) Then we obtain X 1 (w) in Eq. (17a) as X 1 (w) = / , (22a) -ZUJfL(w) -Low--1-gn!l (uj) and L 0 are given by Eqs. (23) and (24). We can also derive x 2 (cjj) in the same way. Appendix B We derive Eq. (34) and show that its denominator has the same form as Eq. (6 28) of LT 3 J except the dipole term. For this purpose we evaluate f(rjjn) = T I: I: Py 2 G (p, w,) G (-p, Wn- IJJv). (B 1) W.v ]J After analytical continuation of iwn from the upper half complex plane, f(cu) 1s reduced to (B 2) we have assumed a particle-hole symmetry. The gap equation can be written in the form (B 3) With Eqs. (B 2) and (B 3), X (co) in Eq. (17) becomes

13 784 H. Takagi and K. Miyake Performing a partial integration with respect to.q, we obtain x(w) as follows: 1/2 x (w) = , -(J) Xorb(w) +g,.+gn (B-4) and The denominator of Eq. (B 4) is cf the same form as Eq. (6 28) of LT. s> References 1) M. C. Cross and P. W. Anderson, Proceedings of the 14th International Conference on Low Temperature Physics, Finland, 1975, ed. M. Krusius and M. Vuorio (North-Holland, Amsterdam, 1975), vol. 1, p ) C. I. Pethick and H. Smith, Phys. Rev. Letters 37 (1976), ) A. ]. Leggett and S. Takagi, Ann. of Phys. 110 (1978), ) P. Walfle, Quantum Statistics and Many-Body Problem, ed. S. B. Tricky, W. P. Kirk and J. W. Dufty (Plenum, New Y ark, 1975), p. 9. 5) D. N. Paulson, M. Krusius and ]. C. Wheatley, Fhys. Rev. Letters 36 (1976), ) K. Nagai, ]. Low Temp. Phys. 36 (1979), ) M. C. Cross, ]. Low Temp. Fhys. 26 (1977), ) H. Mori, Frog. Theor. Fhys. 33 (1965), ) A. J. Leggett and S. Takagi, Ann. of Phys. 106 (1977), ) R. Combescot and H. Ebisawa, Phys. Rev. Letters 33 (1974), ) W. ]. Gully, C. M. Gould, R. C. Richardson and D. M. Lee, ]. Low Temp. Fhys. 24 (1976)' ) K. Maki and H. Ebisawa, Frog. Theor. Fhys. 50 (1973), 1452; Fhys. Rev. Bl2 (1974), 2667.