TEXTURES AND SUPERCURRENTS IN 3He-A

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TEXTURES AND SUPERCURRENTS IN 3He-A

N. Mermin

To cite this version:

N. Mermin. TEXTURES AND SUPERCURRENTS IN 3He-A. Journal de Physique Colloques, 1978,

39 (C6), pp.C6-1283-C6-1288. �10.1051/jphyscol:19786558�. �jpa-00218047�

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C6-1283

TEXTURES

AND

SUPERCURRENTS

IN

3 ~ e - ~ ( * ) N.D. Mermin

Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, N.Y. 14853, U.S.A.

R6sum6.- Cet expos6 presente une revue de rgsultats theoriques r6cents sur la relation entre la texture de l'axe d'anisotropie du paramstre d'ordre de 3 ~ e - ~ et la stabilitg ou l'instabilit6 de cou- rants persistants.

Abstract.- Recent theoretical results are reviewed relating the texture in the anisotropy axis of the order parameter of 3 ~ e - ~ to the stability or instability of persistent currents.

The theory of superfluid flow in 3 ~ e - ~ has acquired some rather novel aspects since LT-14. As a result of these new developments it now appears that superfluidity is a far more intricate, delica- te, and rich phenomenon in 3 ~ e - ~ than it is in superconductors or in superf luid 4 ~ e or 'H~-B / 11.

We now have fairly persuasive theoretical evidence that under a range of conditions wida enough to permit it to be characterized as the normal state of affairs, a sample of 3 ~ e - ~ in absolute thermodynamic equilibrium will support weak but macroscopic persistent currents. This is in contrast to all other known superfluids, in which persistent currents are merely metastable. As if, however, to compensate for this manifestation of a superior degree of superfluidity, larger metastable persistent currents, which can enjoy a remarkable degree of de facto stability in ordinary superfluids and superconductors, appear to have a most tenuous stability in 3 ~ e - ~ . They are prey to many novel mecha- nisms of decay, and, surprisingly, owe whatever me- tastability they do possess to the magnetic inter- actions between the 3 ~ e atomic nuclei.

One way to arrive at these peculiar features of the superflow of 3 ~ e - ~ is to examine the defi- ciencies in a point of view that was widely accep- ted at the time of LT-14 : 3 ~ e - ~ as a superfluid nematic. From this perspective 3 ~ e - ~ is characterized by two independent hydrodynamic fields : a superfluid velocity

ys

analogous to that in ordinary superfluids, and the local direction R of the ani-

-

sotropy axis of the energy gap, analogous to the director in a nematic liquid crystal 121. The flow (*)~u~~orted in part by the National Science Foun- dation under Grant No. DMR 77-18329 and through

free energy density f reflects this structure, being the sum of a conventional nematic term, a conventional (except for the anisotropy in superfluid mass) superfluid term, and an (unconventio- nal) energetic coupling between director and superflow / 3 / :

1 1

=

-

K (a*vx~)~+

-

^{K (V.R)~+}

%(~ExEJ):

fnematic 2 t , - - - 2 s - -

(1)

= - p v 2 1 1

superfluid 2 s s

- T

^p^o⁽^!^'^~^s⁾²^> ⁽²⁾

O V X R

-

co(a~~s)(a~vxa)

.

fcouplingP 1 s

- - - - - -

^{( 3 )}

It follows from this that the mass current (in the v = 0 frame) is :

n

g

= af/avs = psys

-

po R(R=~~)+ C ~

-

RC~R(R-VXR).

- -

^-,

( 4 ) The description provided by (1)-(4) is enti- rely correct. However it is incomplete. Missing is the analogue of the irrotational condition Vxv =O

-

^--S

that expresses in 4 ~ e the fact that the superfluid velocity field is the gradient of the more funda- mental phase 4 of the complex order parameter. Ge- neralizing this condition to '~e-A introduces a further kinematic coupling between v and 11, which

-S

-

provides the analytic basis for an understanding of the curious delicacy of superflow in 3 ~ e - ~ .

As with 'He, the additional condition expresses the structure of the underlying order parameter, whose spatial variation is specified by R and

ys.

In 'H~-A the structure of this order pa-

..

rameter is that of a pair wave-function describing two 'He atoms in a triplet p-state with L = I and S =O /4/. The unit vector field R(r) gives the di-

.-. .-.

rection of the local z-axis in slowly varying non- uniform configurations. Specifying this axis alone does not fully detemine the local order varameter.

the Materials Science Center of Cornell ~niveisi-

since the structure of an L=L = I pair wave-function ty, Technical Report No. 3045.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786558

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C6- 1284 JOURNAL DE : PHYSIQUE

in the relative coordinate p

-

is

$ = (pX+ip If (P) Y

A full specification of the local order parameter is given by the local directions of the x- and y-axes, which I shall denote by the orthogonal unit vector fields @(')(I-)

-

^.., and @(2)(r)

- -

which, of course, determine R

-

~. (r) through

-

(1) (2).

, R = 9 x ? ( 6 )

The order parameter in 3 ~ e - ~ is thus a triad of orthonormal vector fields. Though studied by physi- cists only since 1971, such fields of orthonormal triads have preoccupied engineers since 1909 (the Cosserat continuum) and mathematicians since 1936 (Stiefel manifolds).

The connection between the exotic description in terms of an orthonormal triad and the more in- tuitive description in terms of a director R and a velocity field vs, emerges from the following facts

-

I51 :

(i) A knowledge of the two vector fields R(r) and @;I'(r) (r) is enough to determine,

- ...

by appropriate integration, the value of the full order parameter to within a multiplicative constant ; this pair of fields therefore provides an alterna- tive description ;

(ii) From the behavior of a pair wave-function under Galilean transformation and the fact that a change in phase of the order parameter is equivalent to a rotation of the @(i) in their plane, it follows that the field transforms like 2H/$

times a velocity, leading to the identification

(iii) It can be shown that the condition that an arbitrary vector field v be of the form (7) for

-s

some I$('), @(2) orthonormal and perpendicular to

-

R is precisely

A description in terms of an orthonormal triad can therefore be replaced by a description in terms of R and a field v transforming like a

.., s.-

velocity, provided we recognize that the two fields are not independent, but related by condition (8).

The configuration of R(r)

-

^.., (which I shall refer to as the "texture") therefore affects the mass current in two ways : first, through the ex- plicit R-dependence of the form

-

(4) of g(r),

- -

as was to be expected for a superfluid nematic with

this type of director ; but, in addition, second,

through the implicit dependence of v itself on 2,

-s

-

as required by the additional constraint (8). This latter dependence of superflow upon texture has profound consequences for the superfluidity of

3

He-A

.

Consider, for example, the circulation in v about a closed contour. Because of the relation

-

^s

(8) this need not be quantized as it is in '~e ; it can vary continuously in time provided the texture has the appropriate temporal variation. This sug- gests that it might be possible continuously to degrade a supercurrent without the aid of singular structures (such as vortex lines passing across the current) by suitable variations in a non-singular texture. Since the total mass current (4) is not simply proportional to

vs,

it is not imediatr- ly evident that this suggestion can be realized.

That it can be, is shown by the following example : Let a supercurrent flow around a torus against a uniform texture in which

-

R is everywhere perpendicular to the plane of the torus (which we take to be the x-y plane). If the I$(i)

-

rotate uni- f o m l ~ through an angle 2nn about R

-

as the torus is encircled (figure 1) then equation (7) yields a

Fig. 1 : A configuration in a torus of superflow at uniform _R with one quantum 9 circulation. For m quanta of circulation the $(lf would make m com- plete revolutions about as the torus was encircled.

v of magnitude nH/2Mr, and equation (4) gives a -s

mass current density that is simply psvs. In 'He such a flow would have a high degree of metastabi- lity because of the quantization of changes in phase of the order parameter about closed contours.

In 3 ~ e - ~ , however, we deal not with a phase but an orthonormal triad. Single valuedness of the order parameter does not imply so stringent a quan-

tization condition and the superflow can, in fact, be reduced in a completely continuous mannerwith- out the introduction of singular structures.

To see this we represent the order parameten

(4)

an axis fi and an angle a) the rotation R(ii,a) necessary to take the fixed orthonormal pair 8,y into the local one :

The configuration of figure 1 is described by 6(r)

-

E 2 , while a(r) varies from

-

0 to 2mn as the torus is encircled. A general value of ii and a can be represented as a point in a solid sphere of ra- dius a, the direction from the origin determining 5 and the distance from the origin determining a.

Since rotations through a and -a about the same axis are identical, diametrically opposite surface points must be identified on the sphere representing the possible values of the order parameter / 6 / .

The topology of this order parameter space permits a continuous deformation of the flow pat- tern of figure 1 with a given m into one with m equal to zero or one, depending on whether the ori- ginal m was aven or odd. An example of such a defor mation (from m=3 to m=l) is specified by figure 2,

which shows how to construct a continuous family of paths in order parameter space starting with one representing triply quantized flow at uniform R,

-

and ending with one representing singly quantized flow at uniform

:.

(All configurations specified in the figure are continuous by virtue of the identification of diametrically opposite surface points in the spherical order parameter space).

To appreciate the import of this argument, one must contrast it with the analogous state of affairs in '~e. There the order parameter space is the circumference of a circle (representing the value of the phase) and a flow with quantum number m is represented by a path traversing the circumference m times. No continuous deformation of this path can alter the value of m. To deform away a supercurrent in '~e one must, in a sense, extend the order parameter space. By replacing the circumference of the circle by the entire disc, for example, we make it possible to unwind a superflow at the price of variations in the amplitude as well as the phase of the order parameter. Such devia- tions from local equilibrium (associated, in prac- tice, with the singular cores of vortex lines or rings) are not required to unwind a superflow in 'H~-A, because of the different topology of its order parameter space.

However this melancholy assessment of the

Fig. 2 : A contour encircling the torus provides a mapping of a circle (namely the contour itself) into the order parameter space, each point of the contour going into the point of order parameter

space representing the con£ iguration of the. 3 ~ e - ~ at that part of the contour. The configuration at the top represents three quanta of circulation at uniform _{_R} _;that at the bottom represents a single quantum of circulation with the same uniform texture. The intermediate configurations (with non-uniform textures) provide a continuous deformation of the first configuration into the last.

stability of superflow in 3 ~ e - ~ reckons without one important stabilizing influence : the surface. Sur- face free energies effectively enforce the boundary condition that the $(i) lie in the plane of the

-

surface, i.e. that (1 be parallel to the inward or

-

outward pointing normal /7/. The construction of figure 2 (and, for that matter, the texture we started with) ignores this constraint at the surface, which freezes the texture at the surface, thereby severely curtailing the possibilities for reducing surface circulation in v through varia-

-s

tions in R. Indeed, if

-

.9

-

is required to be plus or minus the outward normal 6 at the surface, then the constraint (8) reduces to the surface relation

where K is the Gaussian curvature of the surface itself. Thus the circulation Qzs.ds in surface

-

contours can change only by the passage across such contours of the singular borders separating

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C6- 1286 JOURNAL DE PHYSIQUE

6

regions of opposite R.L. In the absence of surface tion of the channel.

singularities no continuous deformation of the tex-

A

ture consistent with the boundary condition on 9.

can reduce the surface circulation, and therefore

A

no continuous deformation of 9. that starts and fi- nishes with the same texture can reduce the supercurrent either on the surface or in the bulk.

The presence of surface singularities, however, restores the possibility of reducing the supercurrent by a periodic deformation of the texture. The simplest example of this exploits a surface point singularity, the "boojum", which can be regarded as the limiting form assumed by an island of reversed 8.1 as its area shrinks to a point

/a/.

The boojum (figure 3) has a texture that is non-sin-

Boojums may well be more than the basis for mathematical existence proofs. An analysis of the surface configurations in a container topologically equivalent to a sphere with g handles, reveals that 2 ( 1-g) quanta of surface vorticity are required pu- rely on grounds of continuity, and the realization of chis necessary vorticity by g-I boojums appears to be energetically the most economical way yet proposed

/a/.

(The variety of boojum required for g

2

2, however, is more intricate than the simple circular boojum of figure 3 (see figure 4).

CQ)

Fig. 4 : Two sections of a twisted boojum (proba- bly the more common variety) in planes perpendicular to each other and to the surface.

We thus have a picture of superflow in 3 ~ e - ~ that is painfully sensitive to the motion of surface singularities, whose existence, moreover, is in general a topological necessity. This is not the only aspect of the delicacy of superflow in

He-A. Suppose we consider superflow down a channel in which the surface configuration is constrained Fig. 3 : (a) Cross section of a circular boojum.

~h~ lines are lines of g. (b) A boojum in a super- to be free of all singularities, thereby guaran- current carrying tube. (c) Motion of the boojum teeing the stability of the supercurrent at the around the tube leading to decay of the supercur-

rent. surface. Is this prohibition of surface singulari-

ties enough to guarantee the stability of a uniform gular in the interior of the vessel, but behaves bulk supercurrent such as that depicted in figure like a doubly quantized vortex point singularity ⁵^?

at the surface (with a bending energy that is, how- A conventional bulk stability of such uni- ever, integrable at the singular point). The motion form superflow / 9 / shows that it does indeed repre- of a boojum once around the surface of a channel sent a local minimum of the free energy (1)-(3) if provides an example of a periodic variation in 9. and only if the free energy parameters satisfy the that yields a reduction of the supercurrent by ;wo (co'

iks-~ J)'

R

-

^{< 1 .}

quanta of circulation across the entire cross-sec- OOKb

(11)

(6)

Fig. 5 : Uniform bulk superflow. The stability test is against variations that leave the configuration unaltered at the surface and preserve a boundary condition of macroscopic periodicity along the length of the channel (to simulate a toroidal geo- metry).

Near the transition temperature in the dipole-locked regime we are considering here, the value of R turns out to be 0.9 : uniform superflow just pas- ses the stability test. I find it quite fascinating that dipole locking is essential to this stability.

If the 3 ~ e nuclei had no magnetic moments, so that the spin degrees of freedom of the pairs were not constrained to bend with the orbital ones, then R would have the value 1.5. In this case it is possible to find a continuous family of configurations with monotonically decreasing free energy which

start with uniform bulk superflow and end with a greatly reduced supercurrent confined to the 3ur- face layer (where the supercurrent remains stabili- zed by the constraint against surface singularities). Nuclear magnetism is thus essential to the stability of bulk hydrodynamic superflow in 3 ~ e - ~ !

But nuclear magnetism may not, in fact, be enough to insure such stability. As the temperature drops below T the values of K grows. Cross and Liu /9/ have argued that the threshold (11) is likely to be crossed within the region of stability (against the transition to 3 ~ e - ~ ) of the A-phase.

Takagi _{/ l o /}has also recently pointed out that magnetic fields along the direction of superflow can easily destabilize a uniform flow even in the dipole-locked regime near T

.

Just past threshold there is now some evidence / 1 I/ that the instability does not entail a comple- te collapse of the supercurrent, but a transition to

an intermediate state of flow with a texture that spirals about the direction of a somewhat reduced flow. Our understanding of these possibilities is still quite primitive, and I look forward with some impatience to hearing more about them at LT-16.

Closely related to these theoretical consi- derations, but in a manner that no one has made as

periments of Wheatley /12/ in which driven supercurrents are found to be accompanied by intricate patterns of temporal variation in the texture. It seems likely that these are manifestations of the same topological features of the order parameter space : driving a superflow corresponds to winding the axes p(i) about the local P

-

; to achieve a steady state it is necessary for P itself to turn

.-.

in such a way as to wind down (via equation (8)) the supercurrent as rapidly as the "pinwheeling"

of the

-

is winding it up. There now exists a hydrodynamics 1131 rich enough in principle to des- cribe such phenomena, and preliminary numerical in- vestigations 1141 have revealed very suggestive behavior. Here too, however, our understanding is 'in its infancy and I must again refer you to the proceedings of LT-16 for some real insights.

Having indicated the many ways in which 3 ~ e - ~ turns out to be a very bad superfluid, I should like to conclude with the reassuring obser- vation that in other respects it may be a superfluid par excellence

--

indeed, in a sense, the only genuine superfluid. This is the content of theoretical predictions that in many different geometries the state of absolute thermodynamic equilibrium can sustain macroscopic currents

--

^{a proper-}

ty otherwise enjoyed only on the microscopic level by individual atoms or molecules.

Some simple examples of this are given by configurations in cylindrically symmetric vessels of cylindrical symmetry, where the texture is of the form

The unique non-singular solution to (8) yielding a divergenceless mass current is

which can be viewed as.describing a vortex line that possesses a macroscopic non-singular core.

Such a combination of texture and

-..

yields a non-

.-. S

vanishing macroscopic current denti ty. A simple measure of its size is given by the total angular momen turn,

Near the transition temperature c = p 1 4 , and the integrand is everywhere positive. More remarkably, near zero temperature c = p 1 2 , and the angular mo- mentum is simply H/2 per helium atom, independent

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C6-1288 JOURNAL DE PHYSIQUE

of the actual form of the (cylindrically symnetric) texture. This curiosity cries out for a deeper ex- planation, though I am unaware of any beyond a rather intriguing analysis of the BCS ground state recently given by Takagi and McClure 1151.

The assertion that currents flow in equilibrium in such geometries is subject to the reserva- tion that nobody has proved that the configurations currently regarded as minimizing the free energy

(figure 6) actually do so. No one, however, has co-

Pig. 6 : Various cylindrical geometries in which the configuration of minimum free energy is belie- ved to support macroscopic mass currents : the cy- linder, the sphere, and the lens.

me forth with configurations of lower free energy, though some of the current record holders have been in the literature for almost three years. Once more I look eagerly forward to the illuminations to be provided at LT-16 by further theoretical and expe- rimental studies of equilibrium currents in 3 ~ e - ~ .

References

/ 1 / My own understanding of the phenomena descri-

bed here grew out of two originally unrelated developments : (i) the analytical formulation of quantization of circulation in 3 ~ e - ~ given by Mermin,N.D. and Ho,T.-L., Phys. Rev. Lett.

36 (1976) 594 ; (ii) the study by Anderson,P.

-

W. and Toulouse,G., Phys. Rev. Lett.

2

⁽¹⁹⁷⁷⁾

408 of the implications £or superflow of the topological analysis of Toulouse,G. and Kleman, M., J. Physique Lett.

22

(1976) L-149. I have reviewed in rather more detail some of the points I shall touch on here in my Erice lec- ture notes, published in "Quantum Liquids", ed.

J. Ruvalds and T. Regge (North-Holland) 1978.

121 Except when the contrary is explicitly conside- red, I shall assume that spatial variations are too gentle to prevent (via the additional cost in bending energy) thecomplete alignment with

&

of the axis characterizing the spin degrees

of freedom of the Cooper pair. This so-called

"dipole-locked" regime is the normal state of affairs. I shall also only consider flows in which the normal fluid remains in equilibrium with the walls of the (stationary) container, so that v can be taken to be zero.

n

/ 3 / Other than in a few formulae where it is infor-

mative to display it explicitly, I shall take H/2M to be unity and shall take all free energy parameters to have the dimensions of mass density. The term fCoupling owes its existence to the fact that unlike the director in an ordinary nematic, _R is even under spatial inver- sion and odd under time reversal. Such a term coupling director to velocity field is absent in an ordinary nematic; but this is not the basis for the distinction 1 wish to make.

/4/ The identity of spin and orbital quantization axes is a simplification of the dipole-locked regime, but is not essential to any of the physics discussed here except for the question of bulk stability.

151 Proofs of these assertions are given in my pa- per with Ho cited in reference /I/.

161 Mathematicians call such an object the projec- tive space PB. From the point of view of the topology of order parameter space, the inade- quacy of the superfluid nematic picture can be succinctly expressed : to represent a superfluid nematic one needs the surface of a sphere in 3-space (S2) to specify and a circle (SL) to specify the phase. The order parameter space is thus SLxS2. Although the local structure of SfS, is den tical to the local structure of Pa, g obally they are quite distinct. This is the meaning of the assertion one sometimes encoun-

ters that "changes in the phase" can be defined even though the phase itself cannot.

171 Ambegaokar,V., De Gennes,P.G. and Rainer,D., Phys. Rev.

2

(1974) 2676 and

2

(1975) 345.

181

I have described boojums in some detail at the 1977 Sanibel Symposium. See "Quantum Fluids and Solids", ed. S.B. Trickey, E.D. Adams and J.W.

Dufty (Plenum Press) 1977. See also the article by Anderson,P.W. and Palmer,R.G. in these proceedings, and part 2 of my talk at the 1976 Sussex Symposium, Mermin,N.D., Physica

90

B+C (1977) 1.

/9/ Battacharyya,P., Ho,T.-L. and Mermin,N.D., Phys.

Rev. Lett.

2

(1977.) 1290. A similar analysis has been carried out by Cross,M.C. and Liu,M., Bell Labs preprint.

/lo/ Takagi,S., Tohoku University preprint.

/11/ Fetter,A.S., Phys. Rev. Lett.

40

(1978) 1656 ; Kleinert,H., Lin-Liu,Y.R. and Maki,K., Univer- sity of Southern California preprint.

/ 121 Wheatley,J.C., University of California San Diego preprint. See also his talk at the Sani- be1 Symposium, proceedings cited in reference 7, and also Paulson,D.N., Krusius,M. and Whea- tley,J.C., Phys. Rev. Lett.

2

(1976) 599.

/13/ See for example Ho, Tin-Lun, proceedings of the Sanibel Symposium, cited in refetence 8.

1141 Hook,J.R. and Hall,H.E., Manchester University preprints.

/15/ Takagi,S. and McC1ure.M.. University of Sussex preprint.