Linearized hydrodynamics of 3 He-A1 : correlation functions and hydrodynamic parameters

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Linearized hydrodynamics of 3 He-A1 : correlation functions and hydrodynamic parameters

H. Brand, H. Pleiner

To cite this version:

H. Brand, H. Pleiner. Linearized hydrodynamics of 3 He-A1 : correlation functions and hydrodynamic

parameters. Journal de Physique, 1982, 43 (2), pp.369-380. �10.1051/jphys:01982004302036900�. �jpa-

00209405�

Linearized hydrodynamics ^of 3 He-A1 :

correlation functions and hydrodynamic parameters

H. Brand (*) ^{and H. Pleiner}

Fachbereich Physik, University Essen, ^D-4300 Essen, ^W. Germany

(Rep ^le 22 janvier 1981, ^{révisé le} 27 juillet, accepté le 6 octobre 1981)

Résumé.

Nous présentons ^les équations hydrodynamiques ^linéaires pour la phase A1 ^{de 3He} superfluide qui ^{sont dérivées} à l’aide du formalisme projecteur de Mori tel qu’il ^est appliqué ^à l’hydrodynamique par D. Forster.

A l’inverse des équations hydrodynamiques ^dérivées par des considérations phénoménologiques, le formalisme de projecteur permet ^de produire ^{un contact avec} des théories microscopiques ^{comme la} technique ^{de fonction} de Green. Les équations hydrodynamiques de 3He-A1 ^sont caractérisées, ^au contraire de la phase ^{A et de la} phase ^B

sans un champ magnétique extérieur, par des couplages ^{divers entre} l’espace ^{réel et} l’espace ^de spin, ^{même si} l’énergie dipolaire magnétique ^est négligée. ^En particulier il existe un couplage réversible, instantané entre la densité d’aimantation longitudinale ^et la vitesse superfluide qui produit ^un type d’effet fontaine magnétique ^intro-

duit par M. Liu. La relation de dispersion pour les ^« orbit waves » est présentée pour 3He-A1 pour la première

fois dans une forme correcte et les différences et les analogies ^{de cette} excitation avec les résultats correspondants,

pour la phase ^{A dans un} champ magnétique ^et pour la phase ^{A sans} champ extérieur, ^{sont discutées.}

Nous trouvons également que la vitesse ^et la dissipation ^du quatrième ^son reflètent le couplage réversible de la déviation de la phase ^{et de} l’aimantation longitudinale ^en plus ^du couplage statique qui ^était donné par Pleiner et Graham.

Une liste des relations de Kubo pour les coefficients de transports dissipatifs ^et réversibles est présentée ^{et les}

fonctions de corrélation statiques pour g, 03B4~ ^et 03B4li ^{sont discutées} pour k ~ 0.

Abstract

The complete ^{set of} linearized hydrodynamic equations ^{for the} superfluid A1-phase ^{of 3He is derived}

in the framework of the projector ^formalism by Mori which has been introduced to hydrodynamics by D. Forster.

In contrast to purely phenomenological formulations of hydrodynamic equations ^the projector description

allows to make contact with fully microscopic description like the Green’s function technique. ^The hydrodynamic equations ^of 3He-A1 ^are characterized, contrary ^{to the} A-phase ^{or the} B-phase without external magnetic fields, by ^various couplings ^{of real and} spin space, ^even if the magnetic dipole energy is neglected. Especially ^a reversible, purely instantaneous coupling between the longitudinal magnetization density ^{and the} superfluid velocity ^exists giving ^{rise to the «} magnetic fountain » effect introduced by ^{M. Liu. The correct} dispersion relation for the orbit

waves in 3He-A1 ^is presented ^{for the} first time and the similarities and differences of this excitation when compared

with the corresponding results for the A-phase ⁱⁿ high magnetic fields and for the A-phase without external field

are discussed.

In addition we find that the velocity ^{as well as the} damping of fourth sound reflect the reversible coupling ^of phase

deviation and longitudinal magnetization ^{in addition to} the static coupling previously introduced by ^{H. Pleiner}

and R. Graham.

A list of Kubo relations for the transport parameters, reversible as well as irreversible _ones, is presented ^{and the}

static correlation functions of g, 03B4~ ^and 03B4li ^are discussed for k ~ 0.

Classification

Physics ^Abstracts

67.50

1. Introduction.

Since the experimental ^disco-

very [1-3] and identification [4-6] ^{of the} superfluid,

low temperature phases ^{of 3 He} many papers have

(*) Present address : Institute for Theoretical Physics, University ^of California, ^Santa Barbara, ^CA 93106, ^U.S.A.

been published ^{in this} very interesting, rapidly growing ^field (for ^{a review we refer to refs.} [7-10]).

Most of them deal with the A-phase ^{and the} B-phase

without or in low external magnetic ^fields.

In the present paper ^we give ^a formulation of linearized hydrodynamics of 3He-A1 in the framework of Mori’s projector ^{formalism in order} to connect the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004302036900

phenomenological equations ^derived previously ^with purely microscopic ^{theories on the one hand and}

with experimentally accessible quantities ^{on the}

other. Although there arise various additional hydro- dynamic parameters (static susceptibilities, ^reversible

and irreversible contributions to the currents) ^{it will}

become obvious that ’He-A, ¹ (and ^{3 He-A in} high magnetic fields) ^reveals fascinating properties ^{due to}

the intricate couplings of real and spin space (even

if the tiny magnetic dipole energy is neglected).

The derivation of the hydrodynamic equations given ^{in this} paper is based solely ^{on basic} principles

as thermodynamics ^and symmetry arguments. ^The

specific properties of the considered phase ^results

from the specific form of the equilibrium ^order parameter, AP ^(2.13).

The phenomenological hydrodynamics ^{has been}

derived for various systems in the linear domain _e.g.

for nematics [ 11 ], ^smectics A, B, ^C [ 12], crystals [ 12,13],

cholesterics [14, 15], hexagonal ^discotics [16, 17] spin glasses [18] ^{as well as} in the nonlinear domain e.g.

for superfluid ^4He [19], ferro- and antiferro- magnets [20], ^nematics [21-24], ^smectics A, C and cholesterics [24].

For the superfluid phases ^{of 3 He this} phenomeno- logical approach ^to hydrodynamics ^{has been} pursued

in the linear domain [25-32] and for the nonlinear

regime, taking ^{into account the} nonlinearities which

seem to be most important [33-38].

The use of correlations functions to describe linearized hydrodynamics ^can be traced back to a

paper by ^{L. P.} Kadanoff and P. C. Martin [39] ^who applied ^this technique ^to paramagnets ^and simple

fluids. For superfluid ^{4He the same} approach ^has

been pursued by ^{P. C.} Hohenberg and P. C. Martin [40].

Some years ago this approach ^{has been combined}

with the projector technique ^{of Mori} by ^{D. Forster}

who applied this extended method to the study ^of

nematic liquid crystals [41-43]. Very recently ^the

latter approach has been used to investigate ^the properties ^{of 3 He-A and 3 He-B in low or without}

~~ - -- -

external magnetic ^fields [44, hereafter called I]. ^In

the present paper ^we apply ^this approach to 3 He-Al.

The paper is organized ^as follows : in section 2 we

give ^the microscopic description ^of equilibrium ^i.e.

we present ^the assumptions ^{about the structure of}

the order parameter ⁱⁿ 3 He-A 1 ^{and we} derive the hermitian operators ^{for the variables} characterizing

the broken symmetries and evaluate the commutators

between the latter and the generators of the broken

symmetries.

In section 3 we present ^the linearized equation

for 3He-A, ^{using the} Mori Forster technique. ^The

Kubo formulae for the transport parameters, ^rever- sible as well as irreversible ones are given ^{and the}

static correlations are discussed and compared ^to

those of the A-phase without external field and in

high magnetic ^fields, respectively.

In section 4 the phenomenological linearized hydro- dynamic equations [30, 38] ^are connected with the results of section 3.

In addition we present in section 4-3 detailed discussion of the normal modes which reflect the intricate coupling between real and spin space.

Special emphasis is laid in this section on the _compa- rison of the present results with those of the A-phase

without external field and for the A-phase ⁱⁿ high magnetic ^fields.

In Appendix ^{I we} present ^the gradient ^free energies

for 3He-A, 1 and 3 He-A in high magnetic fields for v"

0 and v" :0 0 and _compare the results with those of BCS-type calculations.

In Appendix ^{II we discuss some} aspects ^{of the} hydrodynamics ^{of 3 He-A in} high magnetic ^fields

and in Appendix ^III fourth sound and entropy (or energy) dissipation ^{of 3 He-A} without external field

are presented explicitly.

2. Microscopic description ^of equilibrium.

^{2 .,1}

HAMILTONIAN AND CONSERVATION LAWS.

When the 3He liquid is considered under the influence of a

magnetic ^{field H} the Hamiltonian takes the form

The magnetic dipole energy has been neglected

because of its tiny ^{order of} magnitude ( 10 -’ K) compared ^with energies relevant for the superfluid phases ^{of 3He} (10-3 K). ^{It can} easily ^be incorporated

into the hydrodynamic equations by ^a procedure

described in [26, 30, ^{31 and} 44] respectively.

In (2.1) tí/:, tfr (X ^{are creation or} annihilation ope- rators for bare ’He atoms which are fermions with

spin ^a

+ -L m ^{is the bare mass and} V(I ^x 1) ^the

bare interaction potential ^{and y the} gyromagnetic

ratio.

The Hamiltonian (2 .1) ^is very useful for a definition of the conserved quantities { Gcx } ^{of the} system ^under consideration. These are characterized by ^{the fact}

that the commutator with the Hamiltonian vanishes.

The conserved quantities ^{are the total} particle number,

the total linear momentum and, if the magnetic dipole energy is neglected, ^the magnetization parallel

to the external field and angular ^momentum.

2.2 ORDER PARA,,~ETER IN THE SUPERFLUID PHASES OF 3He.

In order, ^to establish notation we briefly

sketch the most important ^facts (for ^{a detailed} exposi-

tion cf. (I)). ,

Triplet pairing ^{in the} superfluid phases ^{of 3 He can}

be described by thetnatrix of anomalous expectation

values

where the order parameter Tij is defined by

-

I- i

I

i.e. an integration ^{over the solid} angle of the relative coordinate is performed. ^The normalized matrix Aij

with

characterizes the structure of the condensate whereas the normalization amplitude F(I ^{r 1,} x) ^{is a measure}

for the degree ^of ordering ^not entering ^the macroscopic dynamics ^{of the} system (’).

For the restricted ensemble p ⁱⁿ (2.2) ^{we have}

with where

and /1, v", h, ~io and qaij ^{are Lagrange} parameters (~).

If local thermodynamic equilibrium is assumed to hold (as ^{is the case in the} hydrodynamic regime) ^we

have the Gibbs relation for the change ^{of the} entropy (~) ^{In the} following ^{we assume the structure of the con-}

densate in equilibrium AP ^{to be spatially} uniform thus

restricting ^our considerations to textures with length ^scales

inside the hydrodynamic regime (cf. I).

(~) As usual the ensemble is restricted in the sense that q - 0 only ^{after the} thermodynamic limit has been taken.

where _p is the mass density ^{and s the} entropy per unit mass.

2. 3 STRUCTURE OF THE A1-PHASE (3).

^{In the} At-phase the matrix Aij factorizes [8, 30]

into the complex ^real space ^vector

(like ^{in the} A-phase) and into the complex ^{vector in} spin space di, ^{which has to show the} symmetry ^{of an}

ms

+ 1 or ms

1 state, ^i.e.

Like in the A-phase ^{A defines a} plane orthogonal ^to

the real vector li

In spin space ^{we can} define a unit vector w;

which specifies the direction with respect ^{to which}

ms

1. Because time reversal sends d - d *, w ^{is odd}

under time reversal. It is important ^to note, however,

that the external magnetic ^{field H} already ^{defines a} preferred direction, ^i.e. symmetry ⁱⁿ spin space ^can

no longer ^{be broken} spontaneously ^at T~A ~. ^{In fact w}

orients itself parallel (or antiparallel) ^to the external

magnetic ^{field and can thus not} be considered as an

additional macroscopic parameter. Furthermore it

seems important ^{to note that the} phase ^{of the} complex

vector d is not an independent macroscopic parameter because its changes ^can always be absorbed into 6q~.

Thus the structure (2.8) ^{of the} A1-phase implies

that 3 continuous symmetries ^{are broken} spontane- ously : Gauge invariance and rotational invariance in real _space except for rotations about the axis h ; infinitesimal rotations about the axis li ^{have the} only

effect of changing ^A by ^a phase factor and are equi-

valent to infinitesimal _gauge transformations.

The three real variables _necessary to describe small departures of the matrix Aij(x) ^{from its equili-}

brium value AP(w’ ^{II Z, h° II} y)

(3) ^A corresponding discussion for the A-phase ⁱⁿ high

magnetic fields has recently ^been given by the authors [45].

can be obtained by ^a variation of equation (2. 8) taking ^{into account the} additional conditions (2.9).

Like in the A-phase ^we express 5A~(x) ^{in terms of}

the three real parameters 6q~, blk

with the contraint lil bli ^{= 0. Next we look for 3 her-}

mitian operators 6$ ^and 6( ^with 1;° 6( ^{= 0 in terms}

of 6h; ~ consistent with (2.8), (2.9), (2.10)

The operators ^for b~p ^and bli satisfy ^{a number of} simple

commutation relations

Thus bli ^are scalars in spin space and 6q~ is the infi- nitesimal rotation angle ^{which is} conjugate to M.

Therefore li ^{is a} pseudovector ^{in real} space and b~p

is the infinitesimal rotation angle conjugate ^to lio Li,

i.e. the infinitesimal rotations described by b~p couple spin ^{and real} space. For the behaviour under gauge transformations ^we have

Under time reversal 4ij transforms into 4it ^and Aij

into Ail ^{and thus} equations (2.16), (2.1’~ imply

and 1° ^{- -} 1°. The behaviour of 8l ~ ^and 6$ ^under

Galilei transformations is the same as in the A-phase (cf. D.

3. Hydrodynamic correlations and Kubo formulae in the A1-phase.

^{3.1 THE METHOD.}

^{For the} hydrodynamic description ^{of several} system (e.g. ^{3 He} nonsuperfluid, simple fluids, 4He) it has been _proven to be fruitful to have an approach ^which uses corre-

lation functions, especially ^with respect ^{to the con-} nection between phenomenological ^and purely ^mi- croscopic theories but also if one is interested in experimentally accessible quantities.

D. Forster has made use of Mori’s projector ^for-

malism and applied ^this extended method (correla-

tions combined with projector formalism) ^{to the} hydrodynamics of nematic liquid crystals. Very ^recent- ly this method has been applied ^{to 3 He-A in low or}

without external magnetic field and to 3He-B by

H. Brand, M. Dorfle and R. Graham [44]. ^Because

the general method has been described in those refe-

rences extensively ^we refrain from giving ^{a further}

account of the formalism and refer the interested reader to [42, 43, 44]. However, ^we will list the results for the orbit part ^of 3 He-A without external magnetic

field (without ^detailed derivations) ^{as for as} they ^are

necessary for the understanding ^{of the} hydrodynamics

of ’He-A,.

As it is well known from the general ^{formalism one’}

has to evaluate the matrix of static susceptibilities, ^the frequency ^matrix containing the instantaneous res-

ponse of the hydrodynamic system ^{under considera-} tion and the _memory matrix which consists of an irre- versible as well as of a reversible part, ^both reflecting

the non-instantaneous, collisional contributions of the system. Having evaluated these matrices, ^one is pre-

pared ^to calculate Kubo- and absorptive response functions in the hydrodynamic regime, ^to give ^the dispersion ^{relation of the} hydrodynamic excitations

(appearing ^as poles of the response functions) ^{and to}

obtain sum rules and Kubo relations.

3.2 HYDRODYNAMIC CORRELATIONS AND KUBO

FORMULAE IN THE A 1-PHASE.

^{3 . 2 .1} Symmetries of

the correlation functions ⁱⁿ the A1-phase. ^{- The} , hydrodynamic ^variables { ai I ^{we have to deal with in}

the superfluid A1-phase ^{of ’He are} q, p, 60, 6i; ^and g

where we have introduced the deviation of the entropy density ^{from its} equilibrium ^value

- 0

__o

instead of L(x) [39, 43].

The only additional variable compared ^{to the orbit} part ^{of the} hydrodynamics ^of superfluid ^{’He-A is the} longitudinal magnetization density ^{and it’s therefore}

the main purpose of the present chapter ^to point ^out

the consequences of this fact

Like in (D ^{we introduce the} Cartesian coordinates defined by ^{the three} orthogonal ^{unit vectors}

as a basis in real space. Furthermore ^we decompose

the vectors g and 5t

and introduce

The nine hydrodynamic ^{variables can now be}

divided into four groups according ^to their behaviour under time reversal and spatial ^inversion

As has been pointed ^{out in} (I) this leads to important

consequences for the absorptive response functions because

The symmetry ^relations imply ^{for the} frequency matrix, that its elements vanish, ^if xD is odd in a)

whereas the elements of the matrix of the static sus-

ceptibilities ^{vanish if} xD ^{is even in c~ ;} the elements of the irreversible and the reversible part ^{of the} memory matrix vanish if xD ^{is even} and odd in _w, respectively.

3.2.2 Static susceptibilities ^{in the} At-phase.

For the static susceptibilities ^{of the mass} density ^and

the entropy density ^{we have as usual}

For the inverse static susceptibility ^{of the} longitu-

dinal magnetization density ^{we obtain}

Due to the presence of the unit ^vector w? ⁱⁿ spin

space ^we have two further static susceptibilities ^cou- pling ^the longitudinal magnetization density ^{to mass} density ^and entropy density

Parity and time reversal symmetry would allow

non-zero susceptibilities ^between p, q, p and bit, g2.

However, because the system ^{has axial} symmetry ^for kl ^{= 0} and the correlations of _{p, q, p} and gl, ^~/~

vanish for all k we are forced to the conclusion that

mass density, entropy density ^and density ^of longitu=

dinal magnetization ^{are not} correlated with the other variables statically (it’s, ^{of course,} supposed ^{for arriv-} ing ^{at this} conclusion that _{p, q} and p ^{are not} long

range ordered, ^{so that} x(kl ⁼ 0) ^{= lim} x(k)).

kJ. -+ 0

Thus we can close our discussion of the static sus-

ceptibilities ^of ’He-A, since the static correlations between gl, g2, g3, ^{bit, 61’ and} 6q~ ^are just ^{the same}

as in the A-phase ^{and we refer to} the extensive dis- cussion of these topics in reference [44].

For ’He-A in high magnetic fields, however, ^the situation changes drastically, ^{because one has to} keep

in the list of the hydrodynamic ^{variables one further} quantity characterizing ^a spontaneously ^{broken con-}

tinuous symmetry, bn [45]. ^Under parity ^{and time}

reversal bn transforms like 6q~ ^{and thus, bn} belongs

to the _group of variables named { ay } ^above (eq. (3.6)).

Detailed considerations [46] reveal that the coupling

of the variables gl, g2, g3, 6T, 611, ^{~12 and bn} _gives

rise to the most complicated behaviour of the static

susceptibilities ^{in the} hydrodynamic ^limit (k ^-~ 0) ^of

all superfluids ^{studied so} far. Therefore it becomes

impossible ^to propose experiments ^{which are as} simple and intuitive as those given ⁱⁿ (I).

3.2.3 Frequency ^{matrix and} memory matrix in the

A 1 phase.

The elements of the frequency ^matrix

which couple ^the longitudinal magnetization density

to the other hydrodynamic variables and which vanish

by symmetry ^are c~~~, c~~ p, WIJg1, WIJg2 ^and (o,,6,1- For c~~a~2, WIJg1, WIJg3 ^and c~ua~ ^{we find} by explicit

calculation

The nonvanishing ^of _{c~uy 1, WIlg3} ^and c~~a~ is ^{due to} the fact that there exists a unit vector in spin space which breaks time reversal invariance. The element c~~a~ coupling ^the phase deviation and the longitudinal magnetization may give ^{rise to some kind of} magnetic

fountain effect as has been discussed by ^{M. Liu} [38].

Further nonvanishing elements of the frequency

matrix not coupling to p but contributing ^{to the} hydro- dynamic behaviour of 3He-A, ^are (h/2 m ^{= 1 in}

section 3)

t J - L r~ 1 £’B

For a detailed derivation ^{of this} terms as well for

a discussion why other elements do not contribute to

hydrodynamics ^{we refer to} (I).

Next we turn to the _memory matrix in the A1-phase.

We discuss first the antihermitian part which contri- butes to reversible hydrodynamics and which vanishes

by symmetry ^{if the} corresponding element of the

frequency ^{matrix also vanishes for the same} reason, i.e. we have only ^{to consider} U;g1, U;g3, o~ ^and Q~al2.

The longitudinal magnetization density ^{and the} density ^{of linear momentum both} satisfy a conserva-

tion law, ^thus a" ~ ^and U;g3 ^{do not} contribute to

hydrodynamics ^because c~~91 ^and WIlg3 ^are of lower order in k.

Concerning 6~a~ ^{we can} proceed quite analogous because It ^{satisfies a} conservation law and m ~~~> - k°

thus eliminating Q"a ^from hydrodynamics (~.

Axial symmetry requires ^that

because u"

^{0 and} parity requires ^{an odd} power of k II (5).

The other nonvanishing hydrodynamic ^contribu-

tions of 6~~ ^not coupling ^{to the} longitudinal magnetiza-

tion density ^are (cí I) :

11 11 /. .

1"

The elements of the hermitian part ^{of the} memory matrix vanish by symmetry ^{in the same} cases, in which the static susceptibilities ^also vanish, ^{i.e. the}

elements involving p ^{we have to consider are} ~~P, Q~~, Q~q, Q~g2 ^and ~~’

~ ^vanishes identically because the current of the

density is the linear momentum density, ^a hydrodyna-

mic variable. The latter are projected ^{out when cal-} culating Q~~. ^Thus

~p=0. ^(3.29)

As a consequence of axial symmetry ^for k 1. ^{= 0} ( ~u9, ^{* 0) and} rotational covariance for 7~g(k) ^we

have

For ~u~ll ^{axial symmetry and} parity require

(4) Therefore the magnetic fountain effect is a purely

instantaneous phenomenon ^and picks up ^no collisional contribution. The same holds for the velocity ^{of fourth}

sound as well.

(5) ^{This means the} coefficient _as postulated by ^{H. Pleiner}

and R. Graham does not exist. This can also easily ^{be seen}

in the phenomenological ^framework by ^the consideration of the behaviour under spatial inversion, ^{as will be done in}

section 4.

and for the remaining ^elements involving J.l ^we have

For a complete list of the elements of the irreversible part ^{of the} memory matrix not involving J.1 ^{we refer}

to (I).

3.2.4 Kubo relations in 3He-A,.

As it will become clear from the discussion of the normal modes

of superfluid 3 He-A 1 in section 4, it would be extremely

tedious to calculate Kubo functions and absorptive

response functions explicitly ^{in the} hydrodynamic regime.

Fortunately ^{it is not} necessary ^to do this in order to obtain the Kubo formulae for the transport ^coeffi- cients. As has been shown in (I) ^{one can} proceed directly ^{from an element} of the memory matrix to a

corresponding ^{Kubo formulae.}

Besides the Kubo formulae already presented ⁱⁿ I, equations (5.86)-(5.109) ^{which can be taken over} unchanged ^{to the} A i-phase ^{we find the additional}

Kubo relations

Like in the A-phase without external magnetic ^field

not all sum rules

are satisfied by ^the hydrodynamic expressions ^{for the} absorptive response functions (even ^{to lowest order}

in k) ^{due to} the fact that some of the elements wij(k)

appear in sum with a contribution from a"(k), ^the

reversible part ^{of the} memory matrix.

This leads to the hydrodynamic ^result

For 3He-A1 ^this phenomenon (~) ^{occurs for} x91a~2,

~92~12, Z93~11, X;tg2 ^and x9293 ^{(like in the A-phase without}

external field). Up ^{to now all} hydrodynamic systems

having broken rotational symmetry in ^real space (as

(~) ^{For 3 He-A in} high magnetic ^{fields one further} absorp-

tive response function x~2an, ^{in the} hydrodynamic regime,

violates the sum rule (3.38).

e.g. nematics, ^smectics C, 3He-A1, 3 He-A without, ⁱⁿ

low and in high magnetic fields, ^biaxial nematics [47])

possess those non-instantaneous contributions to the

sum rule (3.38) whereas, ^{to our} knowledge, ^{none of}

all the other hydrodynamic systems, ^not characterized

by ^a broken rotational symmetry in real space (as

e.g. simple fluids, binary mixtures, ^smectics A, ^smec-

tics B, crystals, superfluid 4He, superfluid 3He-B, spin glasses, ^{biaxial discotics} with broken transla- tional symmetries [47]) ^{and studied so} far, ^{show such}

contributions.

4. Linearized hydrodynamics ^of 3He-A1.

^4.1

EQUATIONS ^{OF MOTION.}

As has been shown in [30]

we have to consider as true conserved quantities ⁱⁿ 3He-A1 ^the density p, the entropy density ^{Q, the} density ^{of linear momentum} g and the magnetization density parallel ^to the external magnetic ^field p

(neglecting ^the magnetic dipole energy).

Therefore we have the conservation laws

We refrain from writing ^{down a} conservation law for the density ^of angular ^momentum, which is also a

conserved quantity, because it can be satisfied auto-

.~-- -. - --.- - - - --

matically by choosing ^a symmetric form for the stress tensor (Jij ^{which is} always possible [48, 12, 32]. ^Fur-

thermore we have to keep ^{in our} list of the hydrodyna-

mic variables the quantities characterizing the spon-

taneously ^broken symmetries, i.e. in the case of 3He-A1 phase, ^{we must write down} dynamic equations ^{for the} phase deviations b~p and for the variables characte-

rizing the broken rotational symmetries ⁱⁿ real space

~ : ¹

If local thermodynamic equilibrium is assumed to hold it is possible ^{to use} the Gibbs relation

The equations ^{of state} which relate the intensive variables to the extensive ones remain unchanged compared ^to previous investigations [30].

4.2 REVERSIBLE AND IRREVERSIBLE CURRENTS. -

The elements of the frequency matrix and the anti- hermitian part ^{of the} memory matrix (cf. ^section 3. 2. 3)

relate the reversible parts ^{of the currents} (Eqs. ^(4.1)- (4.6)) ^{with the} thermodynamic conjugates by ^linear

constitutive equations. ^{We have}

where _p is the pressure, mo

xH the equilibrium magnetization,

The cross coupling ^{term -} y between 69 and p ^has

been discussed for the first time by ^Liu only recently [38] (’). ^The irreversible currents take the form as in [30]

and therefore we refrain from writing ^{them down} explicitly.

4 . 3 HYDRODYNAMIC EXCITATIONS IN THE At-PHASE.

- _{For v"} ⁼ ₀ we have to consider six variabies : (’) ^The coefficient _as contained in [30] ^{is zero in order}

to preserve the spatial ^inversion symmetry.

Q, p, b(p, bli ^{and p.} To lowest order in k(c~ ~ k) ^the bli ^are decoupled ^{from the rest} ((1, p, b~p ^and p) ^and

we obtain for fourth sound

where

and and for the two diffusions which are mainly ^{due to} dissipation ^of entropy ^and longitudinal magnetization

where

It is checked easily ^that equations (4.10) ^and (4.11) ^{contain as a} special ^{case the} corresponding results for the A-phase without external magnetic ^{field : if we} put y ⁼ 0, ^{,(,’2 = 0 and} C2 ^{= 0 fourth} sound, entropy ^dissi- pation ^and dissipation ^{of the} longitudinal magnetization ^{of 3 He-A are recovered} (cf. (A. 14) ^and (A. 15)). ^The

additional terms in equations (4. lo) ^and (4.11) reflect the coupling ^of spin- ^and real-space ^{in the} At-phase becoming possible because there exists a unit vector in spin ^space w? which breaks time reversal invariance.

In ’He-A in high magnetic ^{fields the situation} changes completely (due ^{to the existence of the variable} bn,

which characterizes the spontaneously broken rotational symmetry ⁱⁿ spin space) ^{and fourth sound and} longitu-

dinal spin ^{waves are mixed} together [45].

For the orbit waves we find in ’He-A, 1

where

Thus we end up with the striking result that the orbit waves have the identical structure when compared

with the orbit waves in ’He-A without or in low magnetic ^fields.

For the orbit waves in ’He-A in high magnetic ^{fields we have} [45]

where

and

It seems important ^{to note that the orbit wave} frequency ^{in ’He-A in} high magnetic fields has a similar

structure as the orbit ^wave frequency in 3He-A1 ^{and in 3He-A in low or} without external field; ^{it is} only necessary to replace D 2 by C4/ps and, ^of course, it is checked immediately ^that D 2 ^reduces continuously ^to C4/Ps ^for

H - 0.

To arrive at the result for 3He-A1, (4.12), ^{it is} only ^{necessary to} delete all terms involving ^{bn, because this} quantity ^{is not a} hydrodynamic variable in the A1-phase ^{of 3He.}

Furthermore ^we wish to stress that equations (4.10)44.12) ^{have not been} given previously (apart ^from

the velocity ^of fourth sound which has been discussed recently by ^{M. Liu} [38]).

In the case Vn =I ^{0 five variables} (p, s ⁼ alp, 69 and 11 = J1/p) ^are decoupled ^{from the rest to lowest order}

in k. These variables combine to give the first and second sound and ^one mode with m ⁼ 0. Neglecting ^the

static coupling between bT and 6p and between 6h and 6p respectively ^we obtain for first sound (which ^remains isotropic ^{in this} approximation)

where _mo

XH and for second sound

where

Equations (4.15) ^and (4.16) ^{are in} agreement with calculations presented by ^Liu [38]. ^{Without the above}

mentioned approximation ^we get

where

and

For ’He-A in high magnetic ^fields first and second sound are coupled ^to longitudinal spin ^{waves. The}

results for the velocities ^are given ⁱⁿ Appendix ^II reflecting the intricate coupling of real and spin space.

If dissipation ^is included, the modes for Vn =I ⁰

become _very complicated, looking ^{even more} unwieldy

than in case of 3 He-A without external magnetic

field Their main feature is, that m - k for k -~ 0 and dissipation becomes ~ k2 and thus negligible

for sufficiently ^{small k.}

For the remaining ^four variables V x g, bl~ ^reactive

and dissipative parts contribute both to c~ ~ k2 and the corresponding normal modes become extremely complicated ^{for Vn 0 0.}

5. Conclusions.

We have presented ^the complete

set of linearized hydrodynamic equations ^{for the} superfluid A 1-phase ^{of 3 He. These} equations ^show

new and facinating effects which are mainly ^{due to the} couplings ^{of real and} spin space. These couplings ^arise

from all ingredients ^of hydrodynamics : ^{from the}

static susceptibilities and from the reversible as well

as from the irreversible currents (or ^{from the} frequency

matrix and the reversible and the irreversible parts of the memory matrix).

For v" ⁼ 0 we have found that the spectrum ^{of the} orbit waves of 3He-A1 ^{is identical to} that of the A-phase

without external field The velocity ^{and the} damping

of fourth sound as well as the velocities of second and first sound (for ^{v" :0} 0) acquire considerable contributions due to the magnetic ^{field In addition}

to orbit waves and fourth sound we have derived for v" ⁼ 0 two coupled diffusions (c~ ~ iDk2) ^which

are due to the dissipation ^of entropy ^and longitudinal magnetization.

Our formulas are consistent with the sound velo- cities recently given by ^Liu [38]. ^{In the} A-phase ⁱⁿ high magnetic fields the orbit ^wave spectrum c~(k) (for ^{v" =} 0) ^{is similar in its} structure to that without

magnetic ^{fields but is much more} complicated ⁱⁿ

its detailed form; especially ^orbit waves are now

built _up by excitations of bh, 6~o ^{and bn} reflecting

again the intricate coupling ^of spin and real space.

Longitudinal spin ^{waves are mixed} up with first and second sound (Appendix II) ^{for vn =I 0 in a} very

complicated ^{manner due to the} magnetic ^field Therefore, ^{it is} possible ^{e.g. to excite spin waves by}

temperature variations or to excite second sound by ^a fluctuating magnetic ^{field. Because of the} great number of parameters ^{involved and the} complicated anisotropic form of these dispersion relations, ^how-

ever, it seems to be difficult to verify ^all their details

experimentally ^at present.

From the application ^{of the} projector technique

we have found that the reversible current connecting

the longitudinal magnetization density ^{and the} phase deviation, picks up ^no contributions from the collision dominated _memory matrix and is, thus, ^a purely

instantaneous contribution.

Appendix ^I.

Gradient free energies ^of 3He-A 1 and 3He-A ⁱⁿ high magnetic ^fields.

^We present the

expression ^{for the} gradient energy which is implicitly contained in section 3.2.2.

For the A,-phase ^we obtain for v"

0

In general (A. 1) ^{contains six} independent parameters; ⁱⁿ BCS-approximation ^{there are} only three inde-

pendent parameters ^as has been shown _very recently by H. Brand and M. Dorfle [49]. ^For v" :0 ^{0 we} obtain for the A, -phase (cf. ^Section 3.2.2)

Like the orbit part ^{of the} A-phase without external magnetic ^field [32] (A. 2) ^{contains seven} independent susceptibilities py, Pli, ^Ci, Cjj, ^{k1, k2 and k3.} This is due to the fact that the additional hydrodynamic ^variable

does not couple statically ^{to the variables} characterizing the broken symmetries, ^{a situation} completely ^different

from that of the A-phase ⁱⁿ high magnetic field which we discuss now.

For v"

0 we derive from the phenomenological equations of reference [45] ^{the result}

In general (A. 3) ^{contains 11} independent parameters (and ~1 ) ; ^{in BCS} approximation there exist only

three independent coefficients (cf, ^Ref. [49]).

For vn i= 0 we obtain after a straightforward, but somewhat tedious calculations

Appendix ^ll.

Some remarks ^on the linearised hydrodynamics ^{of ’He-A in} high magnetic ^fields.

^The

purpose of the present appendix ^{is the} detailed derivation of the equations ^{of state for ’He-A in} high magnetic

fields and the presentation of the results for the velocities of first sound, ^{second sound and} longitudinal spin

waves.

As is well known [31, 45] ^one has in the A-phase ⁱⁿ high magnetic ^{fields one} additional variable characterizing

a broken symmetry ⁱⁿ spin ^{space bn}

E~~k n) ^{wk 6n;, the component of bni which is} orthogonal ^{to both} preferred directions in spin space (w° ^and n°). ^In [45] ^{we have} given ^the equations ^{of state} involving 6/i., 6(p,

g and bn without derivation. The latter runs as follows.

We choose the following ^Ansatz, which takes into account general symmetry arguments

The requirement ^{that ds must be a} complete differential form leads to the following conditions :

~" ^- ~" ~"~,, j

Combining equations (A. 5)-(A. 9) ^and taking ^{into account that} gi, the density ^{of linear momentum serves}

as both, ^{as a current} (for p)

and as a thermodynamic quantity ^{we have the} additional constraints

Putting together (A . 5)-(A .11) ^we obtain the final result

which coincides with equations (24) of reference [45].

In [45] ^the discussion of the spectrum of the normal modes has been confined to v"

0. For v" :0 ^{0 we} get for the mixture of second sound and spin ^{waves for the sum and} product of the velocities the expressions (neglecting ^static crosscoupling ^{terms between} bp and the other variables)

where

Appendix ^III.

Fourth sound and energy dissipation ^{in 3He-A} without external magnetic ^field.

^Because

the dispersion relations for fourth sound and energy dissipation have, up to now, not displayed anywhere expli- citly ^we give ^{them here.}

For fourth sound the correct results reads

where

and for the _energy dissipation ^{we obtain}

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