Microscopic theory for the orbital hydrodynamics of 3HeA

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Microscopic theory for the orbital hydrodynamics of

3HeA

R. Combescot

To cite this version:

R. Combescot. Microscopic theory for the orbital hydrodynamics of 3HeA. Journal de Physique

Lettres, Edp sciences, 1980, 41 (9), pp.207-211. �10.1051/jphyslet:01980004109020700�. �jpa-00231761�

Microscopic

theory

for the

orbital

_{hydrodynamics}

of

3HeA

R. Combescot

Groupe de Physique des Solides de l’Ecole Normale Supérieure (*), 24 rue Lhomond, 75231 Paris 05, France

(Reçu le 18 decembre 1979, revise le 14 février, accepte le 10 mars ₁₉₈₀₎

Résumé. 2014

Nous _présentons les résultats d’une théorie

_{microscopique}

de la

_dynamique

orbitale de la

_phase

A de l’hélium 3 en _régime linéaire. Des désaccords _importants existent avec les résultats de _{précédentes} théories

phénoménologiques,

mais _également avec certaines relations de

_{l’hydrodynamique}

formelle. Abstract. 2014 A

microscopic

theory is _presented for the orbital _dynamics of 3HeA. There are some

_significant

disagreements with _{purely hydrodynamic} theories as well as with

_{phenomenological}

_approaches. Classification

Physics Abstracts 67.50

Most of the recent interest in

_superfluid

3He

has

been focused on the

hydrodynamic properties

of

3HeA.

This is a

_fascinating

_problem

since

3HeA

is the first

_superfluid

to

_display

orbital

anisotropy [1].

This

_gives

rise to a

_great

deal of

interesting properties

linked to the

_{topological properties}

of the order

parameter

[2].

Of main interest has been also the

related

question

of

_{hydrodynamic stability}

_of super-flow in

_3HeA,

_{where it appears that}

3HeA

could be a rather weak form of

superfluid [3].

In order to treat all these

_{questions properly,}

it is

_naturally

useful to

have a full

_description

of the

dynamics.

With some

exceptions [4],

all the treatments to date have been static ones, where a

_{stability analysis}

of the

_system

has been

_{performed starting}

from the free _energy. But in order to know what

_happens

in unstable

situations,

the full

_{hydrodynamics}

is

_required.

We

will

_actually

restrict our _scope to the orbital

hydro-dynamics

since

_{spin dynamics}

is well understood. A full set of

_equations

for orbital

_{hydrodynamics}

has been derived

_by

Graham

_[5]

and amended

_by

Liu

_[6].

These

_equations

have the

_advantage

of

_being

derived on a

_rigourous

basis. On the other

hand,

they

introduce _many unknown

_parameters

and

actually

are difficult to use in detailed calculations. There is therefore a need for a

microscopic

derivation

of these

_equations

which would

_provide

values for the unknown coefficients.

_Although

several

_attempts

have been

made,

phenomenological

[7]

or

purely

microscopic [8],

a

_satisfactory

derivation does not

exist

_[9].

(*) Laboratoire associe au Centre National de la Recherche

Scientifique.

Here we

_present

the first

_{complete microscopic}

derivation of orbital

_{hydrodynamics.}

The

_resulting

equations

are in

_general

_agreement

with the Graham-Liu

_equations,

and

_satisfy

all the relations due to the

no-growth

of the

_entropy

_by

reactive terms or to

Onsager

relations.

However,

we obtain a non sym-metric stress tensor. Its

_expression

_agrees with the

general

form

_{proposed by}

Hu and Saslow

_[10],

but

not with the relations

_they

obtained

_by

_enforcing

the

angular

momentum conservation law. Our results

disagree

also in some

_important

_respects

with the

phenomenological

treatments

_[7]

on the difficult

points

related to the

_angular

momentum conservation law or the

expression

of the stress tensor. Our

equations provide

also, naturally,

a much

_simpler

form for the

_{hydrodynamics}

because

_they

are

_partly

expressed

in terms of the time derivative of the order

parameter,

rather than in terms of the

_spatial

derivative.

_Finally,

our treatment is restricted to

linear

_{hydrodynamics}

where the order

_parameter

has

_only

small deviations from a

global equilibrium

value. But it

_actually

_gives

an

expression

for

basicajly

all the coefficients

_coming

into non linear

hydro-dynamics

[10],

since

_they

_already

_{appear in linear}

hydrodynamics.

A first

_point

we want to make is that it is natural to

require

that,

in the limit where the

_temperature

T goes to zero, the

_{hydrodynamics}

should reduce to

equations

describing

the motion of the

_superfluid

alone.

This

_implies

that the normal

_velocity

vD should

drop

naturally

out of the

_equations.

This is not an

entirely

obvious condition since some

_transport

coefficients may

actually

not _go to zero because the

relaxation time

_diverges

when T -~ 0. But if this time

L- 208 JOURNAL DE _{PHYSIQUE -} LETTRES is

_kept

constant

_(which

will

_happen

_anyway because

of the finite size of the

_sample),

the normal

_velocity

should

_disappear

at T = 0 since it characterizes the

normal

_fluid,

and there is no normal fluid at T = 0.

This

_property

is satisfied

_by

our

_{equations (this}

is

naturally

a result of the

_theory,

not an

_input).

But it is

not shared

_by

some

phenomenological

theories

[7].

It is also in

_disagreement

with the term

_{(1i/4 m) 1.}

_{curl vn}

introduced

_by

Liu

_[6]

in the rate of

_change

of the

_phase,

since this term does not

_disappear

at T = 0

[20].

This term has been derived on the

_grounds

that the

characteristic vectors of the A

_phase,

_Å1

and

_A2,

have to follow a solid

_body

rotation around an axis

perpendicular

to

_Ai

and

_A2.

But this

_merely

corresponds

to a

_change

in the

_phase

of the order

parameter and,

as in

_{superfluid 4He,}

the

_phase

should not be

_directly

_coupled

to a rotation of the normal fluid.

This term has also been rederived

_{[10] by requiring}

that the

_angular

momentum is conserved. We indeed find an

_angular

momentum conservation

law,

by

combining

the

_equation

of motion for the

_angular

momentum with an intrinsic

_angular

momentum

conservation law. But this does not

_bring

_any further condition on coefficients.

Let us now turn to our results. Our first

_equation

is the intrinsic

_angular

momentum conservation law which

_provides

the

_equation

of motion for

1.

Instead of

_al/at,

we use the instantaneous rotation

_0,

linked to

0110t

by :

allat

= n

x ~

_or n =

I

x

allat

_{since we} are interested

_only

in the _components of n

per-pendicular

to the

_equilibrium

1.

We have :

Here

_Ls

is the

_superfluid

intrinsic

_angular

momentum

_{[11], a}

is the orbital

_{viscosity [12], DEd}

is the

dipole

torque

[1]

and

_{Fi = (2}

_m/~)

_{V~[~/’~/~(V/J] =}

_VjqJij

with Graham’s notations

_[5].

f °

is the free _energy in the reference frame where the normal

_velocity

if is zero.

Explicitly

we have :

where

b2

=

maxk

I A k

12,f’

=

~/7~ where f is

the Fermi

distribution,

r is a relaxation

_time,

_No

the

_density

of

states at the Fermi

surface ; l is

along

the z

_axis,

the other notations are standard.

The tensor _Cijk has the most

_general

form allowed

_by

_symmetry and is

_given

_by

_[13]

_(with

i = x or

_{y) :}

where :

In _eq.

_(4),

_ZJ2

is

_always

_negligible

and has been

_{given only}

for

_consistency.

_X(T)

_goes to

_{pn/4 m}

when T -~ 0

(p

is the mass

_density

and m is the bare mass of

3He).

For T -~

_Tc,

in weak

_coupling.

The second term in _eq.

₍₃₎

is

_completely

new. It is the dominant term in _Cijk near

_T~,

whereas at

low

_temperature

the last two terms take over because of the

_growth

of the relaxation time

T(T,)

when the

tempe-rature is lowered. In _eq.

_(3),

the

_quantity

_b(T)

is

_given

_{by :}

This term is non zero

only

because of

particle-hole

_asymmetry,

but it is of the same order of

_magnitude

as

a(T).

It is difficult to evaluate

_precisely

because

_{No’ EF/No}

is not

_precisely

known,

though

it is

clearly

of order 1.

How-ever, it behaves like

L(T)

T4

at low

_temperature

and is smaller than

a(T)

which _goes like

_i(T)

T2.

On the other

hand,

b(T) -

(1

-

r/FJ

near

_Tc

and dominates over

a(T).

The first and third terms in _{cijk may}

_conveniently

be

_rearranged

with the first two terms of _eq.

₍₁₎

which

can be rewritten as :

constant, all the terms

_containing

VO

_drop

out of _eq.

_(1).

To see this _up to order

_Z~,

one needs the

precise

expression

for the C tensor

_{coming [5]}

in F. In

_particular,

at T =

0,

C 1.

=

p/2

while

CII

= -

p/2

₊ 2

mLS/~C.

Now

_{by inverting-}

_eq.

_(1),

we find an

_expression

for

0110t

which is in

_complete

_agreement with

hydrodynamics

[5].

We obtain for the reactive coefficients :

The result for _{x~ 2013}

_{a2 is}

in _agreement with Graham and Pleiner

_[14],

and Hu and Saslow

_[10].

It is

_easily

deduced from _eq.

_(6).

The

_{approximation}

of

_{neglecting Ls}

is

_always

_very

_good

_except

when one looks at the T -~ 0 limit.

Turning

to the

_dissipative

_{coefficients,}

we have :

Next we consider the stress tensor. We obtain :

where

_c*k

is

_{given by}

_eq.

₍₃₎

_except

that the

_dissipative

terms

_(i.e.

the last two

_terms)

have their

_{sign changed ;}

qJ is the

phase

of the order

parameter

and we have set :

It is clear from _eq.

₍₉₎

that most of the contribution to _(Jij is of

_dynamical

_origin.

-We can see

_that,

because of the first term in _eq.

_(9),

_6~~ is not

_symmetric,

and therefore the

angular

momentum L = r x _g is not

_directly

conserved. More

_precisely,

we have :

where

_cDk

is the

_dissipative

_part

of _ci~k. The first term in the

_right-hand

side is the contribution of the

_superfluid

intrinsic

_angular

momentum to

_SL/~

while the other terms are

_{corresponding}

_quasi

particles

contributions. Now if we add the intrinsic

_angular

momentum conservation law eq.

(1)

to _eq.

_(11),

we obtain a

_good

conservation

law for the

_angular

momentum

_{namely :}

where

_c~

is the reactive

_part

of _cijk.

The_stress

tensor _eq.

₍₉₎

may be transformed if we use the

_equation

_[21]

for the

_phase

_{qJ :}

where

f (T) = f

(dS~/4 lI) dç( - f ’)

(~/E)2 ,

and

_by

=

No

bp

is the fluctuation of the chemical

_potential.

From eq.

(13)

we have

_[15]

for Graham’s

coefficients (

and

_{~~~ :}

In eq.

(13),

as well as in _eq.

_(9),

small thermal terms of order

_(7~/Fp)~

have been

_neglected.

The stress tensor

becomes :

where

_v°p9

=

(p2

i/m2

No)

[ f ~i~ ~pg - f ~ f~pg - f p9 8~~

+

_~.pj,

the

_viscosity

tensor in the absence of

_emotion,

has

_already

been studied

_[16].

We note from _eq.

(15)

that vn

_drops

out when T goes to zero.

L- 210 JOURNAL DE _{PHYSIQUE -} LETTRES

To make contact with

_{hydrodynamics,}

we

_only

need to

_replace

_Qk

in _eq.

_{(15) by}

its value obtained from eq.

(1).

We first obtain for _Uij a contribution :

which _agrees

_exactly

with the

_hydrodynamic

result

_{[10]. (Note}

that it is not

_{symmetrized.)}

Next we have the reactive contribution from

_{V~ :}

where $ =

X(l

+

b/a)

if

Ls

is

neglected

and

Siki

= 4

b~

+

ii

~.

Finally,

for the

dissipative

_part, the

anti-symmetric

contribution from

_cjj~

_Qk

and

_V°pq

_{V pV;}

cancel

_(at

least in the relaxation time

_{approximation)}

and we

obtain :

-where .ae ~

_{(x2/a) -}

_(a

+

_b)2/4

a. Therefore _(Jij is made of the first and the third terms of eq.

(15), plus

the

contributions

_{(16), (17)}

and

_(18).

The reactive and

_dissipative

terms

₍₁₇₎

and

₍₁₈₎

_agree with the

_general

form

_{proposed by}

Hu and Saslow

_[10],

but not with the restrictions

_put

on the reactive coefficients. Indeed we

_have,

_using

their notations :

Finally

we obtain

_{[15] :}

for the thermal

_conductivity

but no term

l

x VT in

the

_entropy

current.

Let us now

_briefly

[17]

outline the

microscopic

derivation. It is a natural

_{generalization}

of a

previous

treatment for the

_homogeneous

case

[11].

We start

with the matrix kinetic

_equation

for the

_quasi

_particle

distribution. Then the motion of the order

parameter

is taken care of

_by

a

_{space-time dependent}

rotation,

which allows one to obtain a scalar kinetic

_equation

for

_{Bogoliubov quasi particles.}

Collisions are then handled

by

a relaxation time

_{approximation,}

though

exact treatment of the collisions are

_{naturally possible.}

However,

in contrast

to

a first

_attempt

[11],

we do not

let the

_{quasi particles}

relax to an

_equilibrium

distri-bution with _energy shifted

_by

the orbital

_Josephson

effect - (~/2 E) 11(k

x

_Vk9O

(CPk

is the

_phase

of the

order

_parameter)

because this shift does not

correspond

to a

_change

in a conserved

_{quantity [18] :}

the orbital

_Josephson

effect is

there,

but

_{quasi particles}

do not relax toward the

_{corresponding}

local

equili-brium. All the

_quantities

are then

_easily

calculated from their

_{microscopic expressions.}

An

_important

point

here is the contribution of the normal

_velocity

to the intrinsic

_angular

momentum conservation law _eq.

_(1).

This is most

_easily

deduced from Galilean invariance.

_But,

_contrary

to the

_{superfluid velocity,}

the normal

_velocity

does not affect the structure of the

quasi particles

and therefore

_only

the contribution

coming

from the

_change

in the

_{quasi particle}

distri-bution should be retained.

Acknowledgments.

- Part of this work has been done

_during

a _very

_pleasant

_stay at the

_University

of

_Sussex,

where I benefited from _very

_stimulating

conversations with A. J.

_Leggett.

I am also _very

grateful

to C. R. Hu for

_stimulating

discussions,

and I thank _very much N. B.

_Kopnin

and K.

_Nagai

for

_sending

me

_preprints

of their work.

References

[1] For a theoretical review, see _LEGGETT, A. J., Rev. Mod. Phys. 47 (1975) 331.

[2] TOULOUSE, G. and _{KLÉMAN, M.,} J. Physique Lett. 37 (1976)

L-149.

VOLOVIK, G. E. and MINEEV, V. P., Pis’ma Zh. Eksp. Teor. Fiz. 24 (1976) 605.

[3] BHATTACHARYYA, P., Ho, T.-L. and MERMIN, N. D., Phys. Rev. Lett. 39 (1977) 1290.

FETTER, A. L., Phys. Rev. Lett. 40 ₍₁₉₇₈₎ 1656.

KLEINERT, H., LIN-LIU, Y. R. and MAKI, K., Proceedings of LT 15, J. Physique Colloq. 39 (1978) C6-59.

SASLOW, W. M. and Hu, C. R., preprint.

[4] HALL, H. E. and HOOK, J. R., J. Phys. C 10 (1977) L-91.

HOOK, J. R., Proceedings of LT 15, J. _{Physique Colloq.} 39

(1978) C6-17.

[6] LIU, M., Phys. Rev. B 13 (1976) 4174.

[7] VOLOVIK, G. E. and MINEEV, V. _P., Zh. _Eksp. Teor. Fiz. 71 (1976) 1129; [Sov. Phys. JETP 44 _{(1976) 591].}

CROSS, M. C., J. Low _{Temp. Phys.} 26 (1977) 165.

[8] KOPNIN, N. B., preprint.

[9] For a review of orbital dynamics, see _BRINKHAM, W. F. and CROSS, M. C., to be _published in _Prog. Low Temp. Phys. [10] Hu, C. R. and SASLOW, W., Phys. Rev. Lett. 38 (1977) 605.

Ho, T.-L., Sanibel Symposium (1977).

LHUILLIER, D., J. Physique Lett. 38 (1977) L-121.

[11] COMBESCOT, R., to be published in _Phys. Rev. B.

[12] CROSS, M. C. and _ANDERSON, P. W., Proceedings of LT 14,

M. Krusius and M. Vuorio, Eds. _{(North-Holland,}

Amsterdam) 1975.

COMBESCOT, R., Phys. Rev. Lett. 35 (1975) 1646.

[13] The last two terms have been found by Volovik and Mineev in their semi _{phenomenological approach,} and also

proposed by Cross in his phenomenological treatment.

[14] GRAHAM, R. and _{PLEINER, H., Phys.} Rev. Lett. 34 ₍₁₉₇₅₎ 792.

[15] This is _agreement with Volovik and Mineev _{(Ref. [7]).}

[16] COMBESCOT, R., Phys. Rev. B 12 (1975) 4839.

[17] The details of this derivation will be _published elsewhere.

[18] This is also in _agreement with a recent treatment of the collision

integral by K. _{Nagaï (preprint).}

[19] They merely relax towards the standard local _equilibrium

corresponding to the shifted chemical _potential (due to the ordinary Josephson effect), the normal _velocity and the shifted temperature. These correspond to the

conser-vation _{of particle} number, momentum and _energy. On the other hand, it is easy to check that a _{quasi particle}

distribution shifted by the orbital _Josephson effect

03B4vk = _{f’ x (2014 03BE/2 E)} 03A9. k x _~k~k does not _correspond

to a _change in particle number, momentum or _energy

because of its _symmetries. Therefore, nothing of it can

survive collisions.

[20] Since the time of this writing, things have somewhat evolved. We have shown _(R. Combescot and T. Dombre, to be

published) that one has to modify non linear

hydrodyna-mics _{[10], by} considering an additional term in the current. At T = _{0, this term} is - (0127/4

m) 013E

x _Vp. This _produces

a contribution in 03BC which cancels exactly the

(0127/4 m) 013E.curl vn term. This brings non linear

hydro-dynamics in agreement with the present theory at T = ₀

with _respect to this term.

[21] There is a _disagreement on this _equation with a _paper on the

same _{problem by Nagai (to} be published). Though both theories agree that vn disappears at T = 0, Nagai finds

an additional term (0127/4 m) Y(T) 013E.curl vn where Y(T)

is the Yoshida function. Since the methods are rather different, it is difficult to know the _origin of the disagree-ment. But it is _probably not _superficial and is rather linked to the approximations made in the theories. It is worthwhile _noting that the term in _question is

quantitati-vely very small anyway, though it is important for the