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Submitted on 1 Jan 1978
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EXISTENCE OF HELICAL TEXTURE AROUND
SUPERFLOW IN 3He-A
H. Kleinert, Y.R . Lin-Liu, K. Maki
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-59
EXISTENCE OF HELICAL TEXTURE AROUND SUPERFLOW IN 3He- A .
H. Kleinert*, Y.R. Lin-Liu, and K.Maki
Department of Physios University of Southern California, Los Angeles, California 90007
Résumé.- On analyse la stabilité d'une texture de superfluide 3He-A en présence d'un courant superfluide J. Se limitant aux configurations dépendantes seulement de z, on trouve trois régimes stables; deux d'entre eux montrent la briseur de la symétrie axiale où le vecteur %
forme une hélice autour du courant.
Abstract.- We analyze the stability of textures in superfluid 3He-A in the presence of a
superflow J. Specializing to purely z dependent configurations, we find three regions in pa-rameter space, which are stable. Two of them exhibit the breakdown of azimuthal symetry where
I is no longer aligned with J but winds around it in a form of helix.
For textures which depend only on z, the direction of superfluid velocity, the free energy of
He-A in the dipole-locked limit can be written as f = j / dz {(pg - pQ cos2B) (az + cosB Yz)2 - 2 cQ
(az + cosB yz) c°s6 sin2B Yz + (^ cos2B+ Ktsin2B)
sin2B Y2 + (K. cos2B + K sin2g) B2} (1)
2 D S z
where p , p etc., are coefficients used by Bhatta-charyya et al. /1.2/. Here I and A are parameteri-z&d 3.s ys
I = (sing C O S Y , sinB sinY, cosB)
A = e-x (- sinY - i cosB cosy, cosy - i cosg siny,
i sinB) (2) Since otis a cyclic coordinate, the z component of
superflow J is completely uniform and can be used to eliminate a from f, by
w- -J <3)
z with g = f - ja = j / dz {B(s) B2 + G(s) Y2 " A(s)-1J2 + 2JH(s) Yz > (4) where A(s) = pjj + PQs, B ( S ) = I ^ (I - s) + Kss, G(s) - [ j ^ (1 - s) + Kts - c2 s (1 - s) A-1 J s H(s) = (1 - cQs A "1) / 1 - s, and s = sin2B (5)The dynamics of % is determined by the Cross-Ander-son equation /3/
- u sin2BYt =(^).-u8t - § | (6)
^Permanent Address : Institut fur Theoretische Physik, Freie Universi.tat Berlin, Arnimallee 3,
1 Berlin.
where y is the orbital viscosity;
We find that there are four types of statio-nary solutions (i.e.
Y. = Bj. -= 0 ) , which satisfy equation (6);
I ( 6 = 0 ) , I I+ (B+ <8 <Bi),
II_(B_ <B <Bi) and III (8z <B < j). The analysis of stability against small oscillations"yields
(compare figure 1 ) .
•8"
/4l-P'
ST ~S S- s 0 0 .2 .4 .6 .8 1.0 tFig. 1 : The three stable regions of the z depen-dent textures in the presence of a superflow are shown.The region I (t parallel to J) is stable onfy for 0 < E < .1. For 0.1 < e < 1, there are two stable regions II+and II_with nonvanishing B and
Y corresponding to helical Stextures. Finally the region III is unstable for all e.
1) I is stable for c > 1 and unstable for c < 1.
2) II and II_ are stable only for c < 1
with Y covering limited ranges depending on
B-3) III is always unstable.
Here
1 "
c =
%
Po (7
Ps + c0r2References
The figure displays the result of our calculation under the simplifying assumption that only p devia-
0
tes from the GL values; 2
(I -€)-I po = :p = cO = - K
5 t,b,s (8) The result (1) confirms earlier analyses /1,4/.The new regions 11+ and 11- correspond to the
?.
vector-
winding around the zaxis in a form of helix. The boundary curves Bland Blare given by
sin2B = I7 + 2E (1
-
E)J
17-
10~](8+~~)-~132
The lower boundaries
B+
(E) and B_ (E) have been calculated numerically. In the limit c -t 1,the stability region shrinks linearly, both
B+
(E)and
8-
(E), converging towards @I(€). In the limit c + 1, the corresponding unique value of stableB(E)
can be calculated exactly without simplifying assump- tion (8) with the result
sin2B = (1
-
c)/(2D) + 0 (1-
c ) ~where
For the values (8), D = + 0 (c
-
1 ) implying sin282
:
(I
-
c).The present work is supported by the Natio- nal Science Foundation under grant No DMR76-21032.
PSI1 Y,
Fig. 2 : The equilibrium positions of
-
in the stability regions I1 and 11- are displayed as functions of sin2f3 or different values of E (E = corresponds to T = Tc, E = 1 to T = 0).Adiabatic cooling proceeds along horizontal lines of fixfd[ps"Jyz].
/I/ Bhattacharyya, P., Ho, T.L., and Mermin, N.D., Phys. Rev. Lett.
2,
(1977) 1290.121 See also, Cross, M.C., J. Los Temp. Phys.
2,
(1975) 525./3/ Cross, M.C. and Anderson, P.W., Proc. 14th Int. Conf. Low Temp. Phys. Otaniemi, Finland, 1975, edited by Kruisius, M., and Vuorio, M., (North Holland, Amsterdam, 1975).
