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Submitted on 1 Jan 1978
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MICROSCOPIC EVALUATION OF THE
TRANSPORT PARAMETERS OF SUPERFLUID 3He
B
Michael Dörfle
To cite this version:
Michael Dörfle.
MICROSCOPIC EVALUATION OF THE TRANSPORT PARAMETERS
JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-28
MICROSCOPIC EVALUATION OF THE TRANSPORT PARAMETERS OF SUPERFLUID 3He B
Michael Dorfle
Universitàt Essen, FB 7, D 45 Essen 1, W. Germany
Résumé.- Les formules de Kubo pour les coefficients de transport de l'hydrodynamique phénoménologi-que de l'He superfluide ont été récemment données. Ici nous calculons ces paramètres dans l'appro-ximation BCS en tenant compte des effets de couplage fort par une fonction de renormalisation Z(u). En particulier, nous discutons la viscosité et la largeur de la raie RMN. Nous comparons nos résul-tats aux résulrésul-tats expérimentaux.
Abstract.- The Kubo formulas of the transport parameters of the phenomenological hydrodynamics of superfluid 3He, which hâve recently been given, are evaluated in the BCS approximation. Strong cou-pling effects are taken into account by renormalization function Z(u). In particular the shear vis-cosity and the NMR line width are discussed and compared with exp e riment s.
We begin with the Kubo relations, which con-nect the dynamic response functions with the trans-port parameters. They bridge the phenomenological théories l\l with the microscopic évaluations of the transport parameters. Shazamanian /2/ used the same procédure to obtain one part of the spin diffusion coefficient. However, the contribution of the order parameter fluctuations to this coefficient was omit-ted in his work.
The superfluid phase is marked by the broken gauge symmetry and the broken rotation invariance of spin space with respect to real space. This requires as additional variables the phase 5$ and a rotation vector in spin space. SQ.. Thèse variables are not conserved, whereas the other hydrodynamic variables (density p, momentum g., energy density e and magne-tization density m.) are conserved.
The example of the viscosity is used to ex-plain the procédure. The momentum is a conserved quantity, therefore
g. + 3. T.. = 0
1 î IJ
T.. is the momentum current density, which follows from the Heisenberg équation of motion. The visco-sity tensor n. .' is obtained from the absorptive part of the dynamic response function of the momentum density by
r, lim lim w x„ ( } ( }
ij a •+• 0 K -*• o k2 • g i . g i
Using the fluctuation dissipation theorem the dyna-mic response function is expressed by the
corréla-tion funccorréla-tion of the currents
-2- X". . (K.to) = ^ - t a n h ^ K . 2 K.„ . (k.oo) k2 g i . g i 2)4(0 2 il m i £ , j mv ' ' (2) Hère, we define ,, /v \ J J » iKr-iu)t KU , j m < K'W ) = d r d t e -< [ TU (r,t), x .m( o , o ) ]+> (3)
We obtain the Kubo formula
3 lim lim „ ,„ . * ^ ,,. \ j = 4 co - o K - o KU , j m "(K'W> K* Km <4>
for the transport parameter. It is useful to define a current-current corrélation function for imagina-ry times
*i*.j»(T> " < TT C TU( 0 , Tj m( f ) ]+> | X = t - f (5) where T is the time ordering operator. The connec-tion between the corrélaconnec-tion funcconnec-tion for real ti-mes and for imaginary titi-mes in the hydrodynamic li-mit takes the form /2/
lim lim Im K.. . (K, ias •*• w + iri) =
ÙJ*O W+o i*,jm
lim lim Ï B w k . , . (K,w) (6) ur*o k-»-o 1 '-|m
Therefore, we hâve to evaluate the following ex-pression
1 lim lim 1 ^ij 4)1 u + o K + o a)
Im K.. . (K, ici) •* w + in) K„ K (7)
îx.,jm J!, m v
T.„ is obtained from the Heisenberg équation of
motion. Then the current-current correlation func- tion will be evaluated for imaginary times and the limites will be taken.
We have the following Hamiltonian
We neglect the weak magnetic dipole-dipole inter- action (%
lo-'
K) which has little influcence on the transport parameters. The Hamiltonian is appro- ximated by the pairing Hamiltonian. The essential strong coupling effects are taken care of by a re- normalization functionThen the renormalized BCS Greens functions read as follows
A is the isotropic gap of the Balian Werthamer sta- te. As third approximation we keep only terms ofthe
A
order 0 (-)
.
E~Starting with the Hamiltonian we get for the momentum current
We factorize the current-current correlation func- tion into one particle Greens functions and get af- ter performing a Fourier transformation with res- pect to the imaginary times
x { ~ r ~+(w,+w~,q) F(wn,q) + Tr ~+(~+w,,q) ~(w,,q)} (1 2) After evaluating the sum over the frequencies and the transition ot the real axis we filially get the viscosity of the momentum. (V(o) is the density of states at the Fermi surface).
Now we want to compare the results with shear vis- cosity measurements. The function Z2(u) is expres-
sed by the function Z (w) in relaxation time appro-
ximation. Then we get
It is interesting to examine the behaviour of
q(T)/r)(Tc) near the critical point. The two fluid
Tc-T
2
lo-b,hydrodynamics is estimated to hold for -5;- and we restrict ourselves to this case. weeobtain
n(~)/q(~~) = 1 + A A2 + B
A2
LnA (15)For T + Tc, the function has a sharp cusp, the
steepness of which is determined by the parameter B.
T -T
For < 1
o - ~
theA2
term with the coefficient ATc
-
is negligible. This result is in contradiction with the results of Pethick, Smith and Bhattacharyya /3/ who predicted (q
-
q)Inc
% -A.Our result is in good agreement with recent and very precise measu-rements of the shear viscosity 141 near T
.
For alarger temperature interval below T we obtain a very good fit to the experimental curve by assuming
Z
a linear dependence of -2 on f3E in the interval
E T
0
5
@- 5 1. The numerical evaluation is also in 2good agreement with the results of a kinetic theory of Einzel und WElfle 151. Besides the shear viscosi-
ty the transport coefficient of the order parameter
-
68. in spin space is accessible to experiment. From
the hydrodynamic theory /I/ follows that the line
1
width of the longitudinal NMR is Aw//=X *2~(~)
Y
where
x
is the -magnetic susceptibility, y the gyro-magnetic ratio and R the shift of the longitudinal
NMR. From our theory follows that V(T) is proportio-
nal to tw
As R2 % A2, we get in the same approximation as
The analogy of this integral to that of the shear viscosity shows that according to our theory the longitudinal line width of the spin resonance must
strongly increase near T
.
Up to now we have no mea-surements of the line width in the B phase in hand which are near to the precision of the viscosity measurements. However, the longitudinal line width of the A phase shows a strong increase near T 1 6 1 .
Similar measurements in the B phase would be very desirable in order to be able to determine the tem- perature dependence of the transport coefficient of the order parameter in spin space.
References
/ I / Brand, H., ~Grfle, M., Graham, R., to be pu-
blished
/ 2 / Shazamanian, M.A., J. Low Temp. Phys.
22
(1976)2 7
/ 3 / Bhattacharyya, P., Pethick, C.J., Smith, H.,
Phys. Rev. B
15
(1977) 3367/ 4 / Reppy, J.D., Phys. Rev. Lett.
40
(1978) 565 /5/ Einzel, D., W&lfle, P., to be published/ 6 / Gully, W.J., J. Low Temp. Phys.
