CHIRAL FIELDS AND 3He

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CHIRAL FIELDS AND 3He

V. Golo, M. Monastyrsky

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-48

CHIRAL FIELDS AND 3He

V.L. Golo and M.I. Monastyrsky

Institute of Theoretical and Experimental Physios, Moscow, U.S.S.R.

Résumé.- Le paramètre d'ordre de 3He superfluide est étudié dans la limite de Ginzburg-Landau, à

l'aide du groupe de symétrie S0(3) x S0(3) x U(l). Nous introduisons des courants hydrodynamiques définis sur une algèbre de Lie, et nous discutons la chiralité. Un exemple de structure en domaines dans la phase B est donné.

Abstract.- In the G-L region we study the spaces of the order parameter for superfluid 3He with the

help of the G_ = S0(3) x S0(3) x U(1) - symmetry. Within this framework we introduce a form of hydro-dynamic currents, which take the values in the Lie algebra LG_, and discuss the problem of chiral theory with several vacuums. An example of domain structure in the B-phase of 3He is indicated.

I.- In this paper we want to point out a few conse-quences of the symmetry of the free energy functio-nal for superfluid 3He in the Ginzburg-Landau

re-gion. This functional depends on a 3x3 complex ma-trix A.. of the order parameter for the p-wave pai-ring and is of the form

F = aTr(AA+) + 0i|TrAA|2 + 32[TrAA+]2 + g3[jr(AA) (AA)8] + e„Tr((AA+)2) + e5Tr((AA)(AA)H) (1)

Here A.. = A... (A*).. = A*., A+ = A*. Functional (1) is invariant under the transformation

A . •* R R. e1<f)A (2)

pi pm in mn

where the repealed indices imply the summation. Transformations (2) constitute the group £ = S0(3) x S0(3) x U(l), where the elements of the subgroup S0(3) (resp. S0(3)2) act in the spin (resp.

orbi-tal) part of the order parameter. The G-invariance of (1) was first pointed out in /I ,2/. In paper/3/ we studied the topology of spaces of the order para-meter by means of the group G^ (cf. also/6/, where we obtain explicit formulae for the order ter) . To restrict the set of possible order parame-ter spaces is important, since the direct methods of investigation of these phases involve minimiza-tion of the free energy funcminimiza-tional. In /2,4,5/ this problem was studied by a combination of analytical and numerical methods. We hope, that the constraints imposed oh these spaces by the symmetry considera-tions of papers/2,6/ together with the analytical methods of /2,5/ will enable us to find all the phases for the p-wave pairing.

Our main idea is, that the phases must be homogeneous spaces G/H, where H is a subgroup of the symmetry group G, /3,6/. We notice, that the similar topological arguments were applied to stu-dy the topologically stable defects and

configura-tions of the ordered media/12/.

Taking into account transformation law (2) we consider the order parameter matrices A with res-pect to their rank and eigenvalues. Hence we obtain the classification of the possible set of phases in the p-wave pairing without any requirements of the unitarity,, the net nuclear magnetization or the ma-gnetic susceptibility. Let A be an order parameter matrix and H(A ) be a subgroup of G which leaves A

o — o invariant. The possible phases are determined by a

choice of A and the subgroup H(A ) , which takes the following values :

1. H(A ) = {l} ; the trivial subgroup ;

2. H(A ) = S0(3), the diagonal subgroup in S0(3) x S0(3), the B-phase is contained in this class ; 3. H(A ) = S0(2) = {(R,R), R belongs to S0(2)} the

unstable 2D-phase is contained in this class ; 4. H ( AQ) = S0(2) x S0(2) = {(Rj.R^, S0(2) j 2 C

S0(3). „} the polar phase is contained in this class ;

5. H ( AQ) = S0(2)i, S 0 ( 2 )iO S0(3)i ( i = 1,2 ;

The A-phase is contained in this class for i = 1.

2.- The G = S0(3) x S0(3) x U(l) - symmetry suggests a definition of superfluid currents with values in the Lie algebra L£ of the group G/3.7/. For the A-phase this was made (unexplicitly) in/8,9/. We specialize only to the orbital dynamics (though the method is quite general and familiar in chiral theo-ries) and define the superfluid velocity as the tensor

vjj = - 3k<|> 6£ j + i(R"1akR)l j, i,j,k, = 1,2,3 where ty and the matrix R are determined by

trans-formation (2) with R = R„, R = 1. We shall not

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write explicit formulae for the currents, (cf. /7/ for the specific information).

We notice, that the conservation laws of mass, momentum and energy imply interesting topological constraints. Let the phases A and B occupy two different regions in a vessel. We suppose that these regions are divided by a domain wall of thickness

5

much smaller than the scale of the regions. Thus we may ignore the structure of the domain wall and

consider it as a surface D. Then we have two maps D+V DtV defined by the order parameter values on

A B

the two sides of D into the phase spaces VA,VB.Hen- ce we have the map W V x VB, its image R is sub-

A

space in V A x VB and it serves to define the hydro- dynamic variables on one side of D by its values on the other side. We obtain the situation similar to the Raleigh curve in gas dynamics.

3.- It is interesting to consider the problem of textures of 3 ~ e within the framework of chiral field theory. We are mainly concerned with textures where solutions of the G-L equations do not remain throughout in the fixed phase space. The reason for this is suggested by the following circumstances. In /lo/ there was found a spherical symmetric solution of the G-L equations, which remains in the B-phase, throughout, but it has a cusp singularity. We have tried to avoid the singularities admitting solutions

the only known solution of this kind. To make sure that the solution is stable we need to study the complex perturbations of fi, i = 1,2,3. This is ra- ther a difficult problem. It is deeply connected with the problem of stability of the phases and heavily relies on the values of the coefficients Bi, i = 1,

...,

5.

In any case our solution suggests, that in superfluid 3 ~ e there may exist textures with several different phases. In this respect superfluid 3 ~ e is the interesting example of chiral field theory with several vacuum manifolds (we notice that the conventional chiral theory deals only with one vacuum). Therefore sophisticated topological situa- tions are likely to appear. One may be reasonably confident, that the multi-soliton solutions familiar in the field theory will be extensively used in the theory of superfluids.

We are thankful to V. Mineev and G. Volovik who draw our attention to this interesting problem, and S.P. Novikov, who pointed out several new oppor- tunities in the field.

References

/I/ De Gennes,P.G., Phys. Lett. (1973) 271 /2/ Barton,G., Moore,M., J. Phys. (1974) 4220 which break away from the phase space. In paper/ll/ /3/ Golo,V., Monastyrsky,M., Preprint ITEP-173 (1976) we have studied a spherically symmetric solution, Ann. Inst. H. Poincar6,

28

n01(1978) 75

which is smooth, iSe. the derivatives of the order 141 Mermin,N-D., Stare9G-3 Report n02186, Cornell University, (1974)

parameter are continuous everywhere, and has the

/5/ Jones,R.B., J. Phys. (1977) 657 form of texture with a domain wall. The domain wall

/6/ Golo,V., Monastyrsky,M., Preprint ITEP-21 (1978) does the B-phase Or any other _{/7/ Golo,V., Monastyrsky,M., Preprint ITEP-16(1978)}

se. It has the form of a spherical layer or a bub-

/8/ Mermin,N.D., Ho,T.L., Phys. Rev. Lett.

36

(1976) ble, inside and outside the bubble there is the B- 594

phase with the order parameter belonging to diffe- 191 Hall,H.E., J. Phys. (1976) L443 rent regions. The thickness of the d-omain wall is /lo/ Mineev,V., Volovik,G., JETP

2

(1977) 767 of the order of the coherence length Rc and R <<R, /11/ Golo,V., Monastyrsky,M., reprint ITEP-127(1977) where R is the radius of the bubble. We assume the /12/ Kl&man,M., Michel,L., Preprint, Univ. Paris Sud

order parameter to be of the form Lab. Phys

.

Solides, ( 1 977)

where fi, i. = 1,2,3, are real, depend on the radius r and satisfy the boundary condition

r = 0 ; f = 0 ; fl,2 = A / n 3

r = c u

,

-

f3 = 0 ; f1,2 =

-

A l n