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Submitted on 1 Jan 1978
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SOUND DISPERSION FOR 3He (B) AND NEUTRAL
SUPERFLUIDS WITH BCS PAIRING AT T = 0
J. Czerwonko
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C 6.9
SOUND DISPERSION FOR
3He (B) AND NEUTRAL SUPERFLUIDS WITH BCS PAIRING AT T = 0
J. Czerwonko
Institute of Physios, Technical University of Wvooiaw, Wybrzeze Wyspianskiego 27, 50-370 Wroeiaw,
Poland.
Résumé.- Dans le cas d'un régime sans collisions et en considérant aussi le terme de l'ordre de kv(kv/2A)2, on a obtenu la dispersion du son pour He(B) et les liquides superfluides neutres avec le couplage BCS pour la température T = 0. On a montré que lorsque seul le couplage est important dans le canal particule-particule, le terme considéré contient les amplitudes spin-direct de Landau limitées à { , = 0 e t à £ = 1. Pour He (B) le terme correctif est négatif pour toutes les pressions. Pour les pressions supérieures à 20 bars le coefficient de ce terme correctif est de l'ordre de quelques milliers.
Abstract.- The dispersion of sound for He(B) and neutral superfluids with BCS pairing is obtained at T = 0 and in the collisionless regime, taking into account also the term of the order of kv(kv/2A)2. It is shown that spin-direct Landau amplitudes restricted to X, = 0 and 1 enter to this term, if only the pairing interaction is important in the particle-particle channel. For 3He (B) the correction term is negative at all pressures. The coefficient near this term is of the order of a few thousands for pressures greater than 20 bars.
The expression for correlation functions of normal Fermi liquids (NFL), in terms of matrix ele-ments of the appropriate resolvent operator, was per-formed by Luttinger and Noziere IM and by Leggett
111. These papers demonstrated that such an expres-sion is useful in some proofs or computations for NFL. We extended this program, 131, into superfluid Fermi liquids (SFL) with isotropic BCS pairing and with' BW pairing /4/. A starting point was Larkin-Migdal equations 15/ for BCS-SFL and our extension of these equations for BW-SFL /6/. The last case corresponds to 3He(B). We restricted ourselves to the consideration of only the pairing harmonic in the particle-particle channel of effective interac-tion and to the collisionless regime. Moreover, only spiniess vertices were considered by us but, on the other hand, our expressions, /3/, were valid also for T # 0.
According to 131, the poles of correlation functions describing spiniess longitudinal response of SFL, are given by B = 0 (BCS), BD - C2 = 0 (BW), (1) where B = <(t2-u2w2)g> - <(t+uw)gRAg(t+uw' )>, (2) C = <(l-w2)(t2-u2w2)g> - <(l-w2)(t+uw)gRAg(t+uw')> (3) D = <(l-w2)g(t2-u2w2-w2)> - <(l-w2)(t+uw)gRAg(t+w') (l-w'2)> (4)
Let us list the symbols appearing above. The bra-cket <...> denotes the average over spherical an-gles, t = u)/2A, u = kv/2A, where co, k are the
fre-quency and the wave vector of an external field, v is the quasiparticle velocity on the Fermi surface and A - the energy gap of SFL. The variable w = cos 9, with 8 being the azimuthal angle. The func-tion g passes, for T = 0, into the funcfunc-tion g(6), B2 = |t2 - u2 w2|, defined in /5,6/. For T 5* 0 g is the Maki-Ebisawa III function, which is not only 3 - dependent. For our present purposes it is suf-ficient to know that, for T = 0, g % 1 + 2(t2 - u2 w2)/3.
Among other symbols appearing in (2)-(4) A denotes spin-direct dimensionless Landau function, defined in a usual way, and determining appropriate Landau parameters. The symbol R is defined as the operator inverse to (1-AQ), where
Q = -j- g(l+?) + uw(l-g)/(t-uw), (5)
and P is the reflection operator of a unit vector connected with the integration over spherical an-gles. Note that the second terms of the formulae (2)-(A) should be treated as the matrix elements of the operator RA between the functions standing on both sides of RA. The multiplication of operators assumes there the average over intermediate sphe-rical angles.
It is easy to see that for u,t « 1 B,C = = 0(u2,t2) whereas D = 0(1). Hence, remembering also (1), one finds that the main terms of the dis-persion of sound coincide for BCS and BW SFL s. In order to calculate the next terms one has to calcu-late B with the accuracy up to fourth order terms
and C,D
-
up to the second order terms. Let us ex-
pand R and g
(6)in (2)-(4)
with respect to t
and u
with the above accuracy. Then, of simple applica-
tion of recurrence properties and the addition the-
orem for spherical functions show us that Landau
parameters fork
=0 and
1enter to such B and C
whereas to D
-
for R
<
3
;D
%
-
2/15 if one ne-
glects second order terms. Finding zeros of the
function B one can express
w
as a function of k
as follows
w
=kv(XY/3)'I2
{I-
(Y-I)(x-I)~
+
$
XY
1,
(6)
9X
'
[
1
where X
-
1+
Ao, Y
Z 1 +Al/3,
with A,
-
Landau
parameters for
9, =0 and
1.The variables X,Y have
to be positive in virtue of stability conditions of
FL 181. The formula (6) gives the sound dispersion
curve for BCS SFL.
For BW SFL the equation (1)
is more complica-
ted
;it contains also A2,A3 via D. Taking into
account that
B =t2/x-u2y/3
+H.O.T. we find that
B
=0(u4) on the dispersion curve. Hence, if we are
looking for zeros of BD-c2, we can neglect second
order terms in D. Hence, taking into account that
C
%2t2/3x-2u2y/15 passes on the dispersion curve
into 4u2/45, and the formula (6) we find.
' 1 2
