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Physica 1OY & 1 IOB (1982) 1615-1628 North-Holland Publishing Company

BROKEN RELATIVE SYMMETRY AND THE HYDRODYNAMICS OF SUPERFLUID 3He

Mario LIU

IFF der Kernforschungsanlage Jiilich, Fed. Rep. Germany

Macroscopic condensed systems sometimes break various continuous symmetries in such a way that they remain invariant under the transformations given by certain combinations of these symmetries. This instance, the breaking of a relative symmetry, is discussed in detail. A system of broken relative symmetry is shown to behave as if it broke all the constituent symmetries but lacked the capability to distinguish between them. Each of the three superfluid phases of ‘He breaks a relative symmetry not encountered in any other known condensed system. Hence each of them is the only representative in nature of a characteristic hydrodynamic response. These symmetries are specified and the corresponding hydrodynamics discussed. In addition, recent work on 3He dynamics and related problems, especially the peculiar thermodynamic properties of 3He, rotating in equilibrium, are reviewed

1. Introduction

Nearly half a century ago, Landau introduced the concept of spontaneously broken symmetry.

During the intensive investigation of superfluid “He in the past decade [I, 31, this idea has once

again proven to be a powerful and unifying tool

in understanding condensed many-body systems.

Below the superfluid transition, 3He simul-

taneously breaks all the continuous symmetries that are broken separately in superfluid 4He,

nematic liquid crystals, and antiferromagnets.

Merely by superposing the various properties of these different systems, one can have a rough idea of the behavior of “He in the millikelvin

regime. Meanwhile. a statement pointing in the reverse

direction can also be made. Detailed study of superfluid ‘He has added a new facet to the

concept of broken symmetry, contributing to a

fuller appreciation of Landau’s idea and enabling us to predict, and understand, collective behavior unsuspected until recently.

This new facet is the breaking of a relative

symmetry. Generally speaking, given two continuous symmetries, X and Y, it is not difficult to

conceive a state which is invariant under the

transformation given by a certain linear com-

bination of these two symmetry operations, but is not invariant under any deviation from this

linear combination, especially X or Y alone.

The intriguing point here is that this state will

respond in ways characteristic of a system breaking both X and Y, while mistaking one for

the other. For example, the ‘He-B phase breaks a rela-

tive spin-orbit symmetry [I] that is a linear

combination between rotations in spin and orbital space. Its dynamics, therefore. includes

aspects of antiferromagnets [4] and of nematic liquid crystals [5]. At the same time, its response to mechanical rotations is, within certain limits,

indistinguishable from that to magnetic lields,

representing spin space rotations. In other words, in the B-phase we can mechanically

generate typical antiferromagnetic responses.

spin waves and NMR, or, vice versa. magnetically induce nematic shear instabilities [6].

The A-phase of superfluid ‘He breaks the relative gauge-orbit symmetry (GOS). It is characterized by the equivalence between gauge transformation and a certain orbital rotation.

0378-4~~~/~2/0000-0000/$02.75 @ 1982 North-Holland

1616 M. Liu i Broken relative symmetry of superfluid ‘He

Hence, in addition to being both superfluid and

nematic [4,7], the A-phase provides the pos-

sibility of changing the phase by mechanical

rotations. A little disk, immersed in helium and

turned at a constant angular velocity, will wind

up the phase and generate a superfluid velocity. This is balanced by a counter normal current,

with the direction of heat transfer depending on

the sense of rotation. The temperature of the

region close to the disk can thus be changed at

will. This disk was called a “gauge wheel” [S], emphasizing the existence of a handle with which

to crank the phase mechanically.

Fortunate enough for helium physicists, the

magnetic Al-phase breaks yet another relative

symmetry, the spin-orbit-gauge symmetry (SOGS). This time, it is a linear combination of three continuous symmetries. So the response of

the Al-phase to magnetic fields, mechanical rotations, and phase windings is, on the one hand, very similar, but on the other hand,

modelled after antiferromagnets, nematic liquid

crystals, and superfluids, respectively. In addition

to the features already expected of the A and

B-phases, the occurrence of a magnetically

generated superflow may be taken as Al’s

fingerprint [9]. Recently, Corruccini and

Osheroff investigated the Goldstone mode, a spin-temperature wave, of broken SOGS [lo].

This is the first time that any direct consequences

of a broken relative symmetry have been detec-

ted, and the agreement between theory and experiment is quantitative. Incidentally, the experiment has also revealed the magnetic

quantum number of the pairing state, a piece of information of evidently macroscopic quantum

nature. In the next section I shall elaborate on the basic

ideas underlying the hydrodynamic theory and work out some points (in subsection 2.1) not stated explicitly before. They are quite useful for the discussion in section 4 of the ther- modynamics of superfluid 3He, rotating in equilibrium. In section 2, I shall also introduce the idea of broken relative symmetry with the help of a

simple, two-dimensional model. Chapter 3 con-

tains a study of various superfluid phases of “He

employing this idea. In section 4, rotating equili-

brium is examined and recent work on 3He

dynamics reviewed. Consistency between micro-

scopic results and the hydrodynamic expressions

containing only the lowest order, local equilibrium terms is demonstrated. Section 5 con-

cludes this paper with a brief summary.

2. The hydrodynamic theory

The hydrodynamic theory describes the static

and low frequency dynamic properties of a

macroscopic system. With only a handful of

variables, it is an extremely simple yet rigorous

theory. The essential input of the theory is the information about the system’s symmetry, with the continuous ones assuming a dominant role.

Any two systems obey hydrodynamic equations of identical structure, if their Hamiltonians dis-

play the same continuous symmetries and if an

equal subgroup of these symmetries is spon-

taneously broken. Both have the same variables and are characterized by an equal number of

propagating and diffusive modes. In comparison, the discrete symmetries are less consequential.

They determine the number of independent elastic and transport coefficients and are thus

relevant to the question whether certain modes are coupled, or others are degenerate. The dynamics of solids [ 11, 121 serves as a well known example here. Any solid has, in addition to the conserved quantities, the displacement vector as

three extra hydrodynamic variables to account

for the broken translational symmetries; and irrespective of the crystal point group, the col-

lective modes are three pairs of elastic waves, a heat and a defect diffusion [13]. However, due to

the difference in the discrete symmetries, glass has two independent elastic constants, while a triclinic crystal has 21.

From these considerations we observe that the hydrodynamic theory provides a very convenient

M. Liu / Broken relative symmetry of superfIuid ‘He 1617

scheme to classify condensed systems. With a

dozen or so continuous symmetries to break in various combinations (not all of which are possible), there are only a rather limited number of

different basic structures a hydrodynamic theory

can have. These theories cover the whole range of low frequency response of all conceivable

macroscopic systems.

2.1. Local equilibrium and gradient expansion

The hydrodynamic variables H;: are connected

in a rigorous fashion to the system’s symmetry.

They are either locally conserved quantities, such

as the density p, or variables of continuous

symmetry transformations, such as the macroscopic phase variable C$ in 4He-II. The conserved

quantities reflect the continuous symmetries of

the interaction, while a symmetry variable appears when the state breaks the corresponding

symmetry spontaneously [14]. In contrast to the vast majority of the 10z3 degrees of freedom of a

macroscopic system, the hydrodynamic variables possess characteristic times that diverge in the

limit of vanishing wave vector q. When dis-

turbed, the system restores equilibrium by transporting the mass, or transmitting the in-

formation about the phase variable’s average

value. The times 7? needed for these processes

obviously diverge with the wavelength of the disturbances. All other microscopic degrees of freedom F, relax, in this limit, with constant,

non-vanishing rF [1.5]. As long as we restrict q to

be such that for all i, j: T?(q)% T/F, there is a frequency range. ~6’ 9 T:, called the hydrodynamic regime, in which the hydrodynamic variables alone determine the dynamics of the

system, while all the E; instantaneously assume their static values, determined by the H,. Since there is no reason to always expect a local relationship, we have in a gradient expansion

fi(t, r) = E(H,(t, r), V,H,(t, r), . . . ) . (1)

In global equilibrium, the H, are usually constant

and eq. (1) reduces to F, = Fr’(Hj). One of these Fi is the entropy density s, and the maximization

of the total entropy (with appropriate con-

straints) yields dS/dHi = const., leading to the familiar equilibrium conditions. such as given by

the constancy of temperature T, chemical potential p, velocity 2), or of vorticity fl = $V X 2) in a

rotating system. (The broken symmetry variables behave slightly differently from the conserved Hi considered here. We shall examine them

separately below.) For finite but small frequencies, it is vastly simplifying and usually sufficient

to assume local equilibrium, i.e. to retain only the

leading order terms of eq. (1):

F,(t, r) = Fyq(H,(t, r)) . (2)

Generally, however, the hydrodynamic vari-

ables are not uniform in global equilibrium: consider the density of a rotating system or in

the presence of non-uniform textures. Then the

zeroth-order terms may fall short of local equilibrium, while a higher order gradient expansion will in general yield bona fide hydrodynamic

terms in the same order of q as those going

beyond the local equilibrium approximation, forcing one to abandon this attractive and phy-

sically plausible concept. An elegant way to cir-

cumvent this problem is to take the F, as func-

tions of the thermodynamically conjugate variables, such as T and p, now defined as the appropriate functional derivatives &/6H,. Because these variables are by construction uniform in equilibrium, Fpq cannot contain any

gradients of them. To put it technically, because V,u is zero in equilibrium, it should be smaller

than Vp for hydrodynamic frequencies and hence represents a more effective expansion parameter. We conclude that the content of locafi equili-

brium, assumed here throughout as well as in virtually all works on hydrodynamics, can be taken quite generally as the restriction to the zeroth-order terms in an expansion of F, with respect to the gradients of the conjugate variables, or more physically. with respect to devia-

1618 M. Liu / Broken relative symmetry of superfluid ‘He

tions from equilibrium. It is of course by no means necessary for Fi to depend on VHj, if

these are non-vanishing in equilibrium. For in-

stance, in the case where the density is in-

homogeneous, the usual reason is either the non- uniform distribution of angular momentum den-

sity in rotating equilibrium or a spatially depen-

dent texture, both of which modify the density through thermodynamic cross derivatives. In the

absence of these disturbances, p is usually constant. Here, a local relationship p = k@) - in-

stead of /.L = p (p, Vp) - or equivalently s = s@),

is obviously an adequate description, and the

local equilibrium form of the entropy density consists of zeroth-order terms in both types of

variables. The entropy production function R restores

global equilibrium from the local one. It serves

as a potential and, by seeking to reduce itself, forces the hydrodynamic variables toward equil-

ibrium. In equilibrium, R vanishes identically. In more general circumstances, it is positive

definite. Hence, in an expansion of R, again with

respect to the gradients of the conjugate variables, the lowest order terms are of second order, given by (ViT)*, ViT(Vjuj + Vjvi). . . . This

leads naturally to the proportionality between the fluxes of the hydrodynamic variables and

their respective generalized forces.

We observe that the standard forms of both s and R contain only the leading order terms. A

further expansion is possible but does not appear

rewarding, since (i) the next order corrections, likely to be of any significance only for high q-values, have to be dearly paid for with a proli-

feration of elastic and transport coefficients; and (ii) the analyticity of s and R is far from established, and the next order terms may well be smaller than the non-analytic ones.

The inclusion of broken symmetry variables only slightly complicates the arguments. It does, however, further diminish the information yield from indiscriminately counting the number of gradients. Taking the angle of the infinitesimal rotation 0” in a biaxial nematic liquid as an

example [16], the elastic energy is obtained by

keeping the positive definite leading order term

-V,t?~V& in a gradient expansion. The ad-

ditional variable, Vie:, is hence of the same order as p or U. The molecular field, defined as vi =

TS$s dvol/SBY, vanishes in equilibrium and is of the same order as VT or VP. The entropy production function R hence contains lyf, pos- sibly even ?PiViT. However, !Pi cannot be written

as the derivative of some potential, but is rather

a sum of several expressions, each of them containing two gradients but none of which is

necessarily zero in equilibrium. Therefore, they

are individually likely to be larger than, say, VT

or VP. In general, 0: is non-uniform, and so is

the local orientation of the symmetry axes. This

contributes to the spatial dependence, even in global equilibrium, of the hydrodynamic vari-

ables, but still leaves the conjugate variables

constant. It is confusion about one or the other point in

the above discussion, I believe, that is to a large extent responsible for the controversies concern-

ing ‘He-dynamics, intrinsic angular momentum,

and rotating equilibrium. Section 4 contains a

simple derivation, with the help of which I have convinced myself that all terms of interest are of the same and lowest order in a proper hydrodynamic expansion, and hence lie within the

approximation of local equilibrium. In what follows, we shall examine a few well-

known examples, starting with water, then going

over to antiferromagnets, nematic liquid crystals, superfluids, and crystals, each representing a

different type of broken symmetry.

2.2. Systems without a broken symmetry

Any neutral, single component system con- serves energy, momentum, angular momentum, and mass, reflecting the invariance of its internal interaction under the infinitesimal transformations of translation in time and space, rotation and gauge transformation, respectively. An additional invariance under the rotation of the spin

M. Liu I Broken relative symmetry of superfluid ‘He 1hlQ

degrees of freedom alone adds the total spin to the list of conserved quantities. If the equilibrium state of the system displays the same

invariance. i.e. if it does not break any sym-

metries of its interaction, we may refer to it as a paramagnetic normal fluid. Its hydrodynamic

variables are given by the eight densities of

energy E, mass p, momentum g, and spin S.

(Instead of the energy, it is more convenient to take the entropy s as an independent variable, as

I shall do from now on.)

The eight collective modes of this fluid are a

pair of sound waves, and six diffusive modes. As

is easy to realize from the hydrodynamic equations [17]. the existence of the propagating modes is

intimately related to the hydrodynamic velocity

v, which represents the only reactive and hence effective transport mechanism in the system. It is also universal, carrying all the conserved quantities at the same time. In contrast, the various

diffusive currents are both specific and slow. They transport one conserved quantity each and

are necessarily dissipative [ 181.

In the next subsection, we shall study the hydrodynamics of systems with various broken sym-

metries. Their common outstanding feature will

be the appearance of an effective and specific

transport mechanism - for that conserved quan-

tity whose corresponding symmetry is broken. As in the case of u, this new effective means of

transport usually leads to a propagating behavior

of its previously diffusive conserved quantity.

2.3. Systems with one type of broken symmetry

If the paramagnetic normal fluid undergoes a transition and breaks a continuous symmetry in its ordered phase, the variable of transformation

of this symmetry provides an additional hydrodynamic variable, changing the hydrodynamic theory in two ways. First, its equation of motion is to be included in the set of hydrodynamic equations, and second, this variable couples to the existing ones and alters their equations of motion. (I shall restrict the discussion to the

equilibrium terms only, i.e. to those terms that

remain non-vanishing in the limit of zero entropy

production, R = 0.) The essential modification

occurs in the equation of the conserved quantity

that is conjugate to the broken symmetry. and

this modification can be interpreted as a neu

current - specific, dissipationless. and proportional to the gradient of the symmetry variable.

Let us first examine the antiferromagneric

normal fluid, a liquid that spontaneously breaks the rotational symmetry in spin space. The rota-

tion angle 0; enters the hydrodynamic theory as an

extra variable, providing an additional spin current

-Vet. This, in turn, converts the spin diffusion

into a spin wave, the Goldstone mode of broken

spin symmetry. The additional equation of motion

(3

shows clearly that the spin angular velocity w, is

a field of special relevance to antiferromagnets.

Other systems, as we shall see. have different

relevant fields. Note that didt = ali% + UV is the

material derivative and w, is defined such that all

the other hydrodynamic variables are held con-

stant when the derivative is taken. Usually, w, =

~(J&/x-- H,), and the relevant tield can be

thought of as a high frequency magnetic field yAH, varying on a time scale much smaller than

the longitudinal relaxation time. Our next system is the supe$uid which breaks

gauge invariance. (For the sake of simple arguments and because a paramagnetic superfluid is not known to exist, I shall neglect the spin

degrees of freedom here.) In this system, the

gradient of the macroscopic phase variable, VP, provides an independent mass current. In con-

trast to the universal transport via u (denoted as

o, in the literature on superfluids). Vrp carries

nothing but mass, and especially no entropy [ 171. Being dissipationless, this new mass current need not vanish in equilibrium, and a sufficiently small current persists indefinitely, leading to the name of superfluidity. A persistent, equilibrium mass current, being at the same time a momentum

1620 M. Liu / Broken relative symmetry of superfiuid ‘He

density, is itself a hydrodynamic variable and hence detectable. Due to this circumstance, it

was deemed more spectacular than the formally

equivalent persistency of spin or momentum

currents.

The Goldstone mode of superfluids is second sound, a temperature wave. The apparent mis-

match between the appearance of a new current

for mass and the conversion from diffusive to

propagating behavior of the entropy can be

resolved quite easily by employing the Galilean

covariance principle. The persistent mass cur-

rent, when viewed from an appropriately chosen

inertial system, appears as a persistent entropy current. And this can be taken as what effects

the conversion. Note, however, that the walls are moving in the second inertial system and the above equivalence can only be correct for bulk.

In fact, by creating an abundance of boundaries,

such as in a confined geometry, we do alter the nature of the Goldstone mode and observe in-

stead fourth sound, a true density wave. The lattice of a hypothetical superfluid crystal also

provides a preferred inertial system. Here, a

persistent current of mass is inherently different

from one of entropy. The first would lead to the propagation of defect density, and the other to a

temperature wave [19]. The equation of motion

of cp is the Josephson equation

dqldt = -2mplh, (4)

with the chemical potential now being the relevant field, acquiring the same significance as the

spin angular velocity (or the high frequency magnetic field) in antiferromagnets. The chemical potential SE/@ is again evaluated by holding constant all the other hydrodynamic variables.

Our third example, the nematic liquid crystal, breaks rotational symmetry in orbital space. The gradient of the rotation angle Vi07 enters the equation of angular momentum conservation as a torque or, equivalently, as an angular momentum current [5, 121. In the general case of biaxial nematics, the Goldstone modes are two pairs of

orbital waves and one orbital diffusion. If the

longitudinal and transverse variables decouple,

as they do for certain directions of the wave

vector, it is the longitudinal angle (denoting the

rotation around the wave vector) that diffuses,

while the transverse angles and velocities join to form the modes of orbital wave [16].

The equation of motion for 8” is given as

dO”/dt = 0, (5)

where the vorticity 9 = 40 x o, or the system’s

response to a change of the angular momentum,

now represents the relevant field. It is the nema-

tic counterpart to the chemical potential or the magnetic field. A clear distinction between LZi

and vi, = i (Vjv, + Vjvi) is of crucial importance, although both are of the same order in 4. The

former is an equilibrium field, the latter is a

generalized force and a measure of deviation

from equilibrium. For instance, it is incorrect to expand R in V,vi, as is frequently done.

Finally, we have crystals as the best known

example of systems that break translational

symmetry. The displacement vector u is the symmetry variable of translation, while ViUj,

proportional to the stress, can be interpreted as an independent momentum current. With duldt = v, the relevant field is obviously the velocity, and the Goldstone modes are two pairs

of shear waves and a defect diffusion [12]. Except for translation in time, these examples

account for all the continuous symmetries that

can be broken in a macrostate. The problem we are now going to address is the coexistence of

different types of broken symmetries.

2.4. Broken relative symmetry

Instead of tackling the problem of coexistence in its whole complexity, we shall examine a simple model liquid, in two dimensions, with only one spin degree of freedom and a phase variable, depicted as arrows and clubs, respectively, in figs. l(a-e). Fig. l(a) is meant to represent a

M. Liu / Broken relative symmetry of superj7uid jHe 1621

Fig. 1

system invariant under both rotations and gauge transformations. In other words, if the figure

were bigger, with many more then four points,

and we would lose track of the individual points,

then a rotation of all arrows, or clubs, by a

certain angle would leave the picture of chaos

unchanged. it is a paramagnetic normal fluid. In

fig. l(b) the spin-arrows are ordered and a rotation would lead to a perceptible change. This is

what we may call an antiferromagnetic normal fluid. Fig. l(c), a paramagnetic superfluid, dis-

plays an ordering instead in the phase clubs. Fig.

l(d), the tidiest one, breaks both rotational and gauge invariance. It is an antiferromagnefic superfluid. In this state, both 8” and cp, together with their equations of motion, enter the hydro-

dynamic theory, providing an independent mass current and an extra spin flux. These, in turn,

give rise to second sound and spin waves. Given two degrees of freedom, the concept of broken symmetry in our straightforward interpretation yields only four basic types of hydrodynamic theories.

This conclusion is. however, incorrect. And

the error stems from the implicit assumption that

a symmetry is either broken or not. Consider fig.

l(e)! It is obtained by disturbing fig. l(d) while

maintaining the constant angle between the arrows and the clubs. The symmetry broken is

obviously the relative orientation or, in current

usage, a relative spin-gauge symmetry. How does

such a system behave? We observe that both a spin rotation and a gauge transformation change

the relative angle and alter the state perceptibly.

This distinguishes fig. l(e) from figs. l(a)-l(c) and implies both antiferromagnetic and

superfluid behavior. On the other hand, once the angle is changed, we have no way of telling

which of the two operations was performed. i.e.

a system of broken relative symmetry mixes up

the two constituent symmetry transformations. This is the crucial difference to fig. l(d)., the true

antiferromagnetic superfluid. The broken rela-

tive spin-gauge symmetry requires the linear combination cp - 0” as the additional hydro-

dynamic variable, measuring the change in the angle after both transformations have been per-

formed. The new current, provided by V(p - P),

carries spin and mass at the same time. In other

words, this quantity enters both the continuity

equation and the equation of spin conservation. The Goldstone mode of relative spin-gauge

symmetry is therefore a propagating wave, in

which both spin and temperature participate. Finally, the equation of motion. given by com-

bining eqs. (3) and (3), is

d(cp - O’)ldt + 2mplh + o = 0. (f3

and the confusion between the symmetry operations leads to one between the two relevant physical fields.

One direct consequence of this equation is the

magnetic fountain effect. Take two vessels con-

taining our model liquid which are connected by a superleak to suppress any exchange via u,. if a magnetic field is suddenly turned on. or increased, in one of the vessels by yilH, the system will respond, with the help of the non-equilibrium terms neglected here, in such a way that

1622 M. Liu I Broken relative symmetry of superjluid ‘He

d(cp - 8”)ldt quickly vanishes, with the difference

in the spin angular velocity between the two

vessels balanced only by the difference in the chemical potential Aw = -2mA/~/h. Neglecting the three small quantities of magnetostriction,

relative magnetization, and the squared velocity

ratio of spin wave to fourth sound, we have

do = -yAH, leading to a magnetically induced

pressure change. The reverse effect, of course. is also possible.

The indistinguishability between a phase

transformation and a rotation, by the way, is a

familiar property of any quantum mechanical

system that is an eigenstate of $,, with a non-

vanishing magnetic quantum number M,. Rotat- ing the wave function by 8” around i with a subsequent gauge transformation of cp yields the

factor exp i(cp -MS@). If this is the state a system Bose-condenses into, the wave function’s (rela-

tive) symmetry becomes a macroscopic property.

And the fountain effect would yield the beautiful macro-quantum relation

Ap/Ao = M; h/h (7)

Summarizing, we observe that condensed sys-

tems may spontaneously break a linear combination of two continuous symmetries. The

response of a system breaking this relative

symmetry is characteristic of one that breaks

both constituent symmetries but lacks the ability to distinguish between the corresponding rele-

vant fields. Equipped with this heuristic principle

and some simple knowledge, such as presented in subsection 2.3, about the hydrodynamic

behavior of the four basic systems, it is quite

painless to achieve a thorough qualitative grasp of the complex hydrodynamics of superfluid 3He.

3. Hydrodynamics of superfluid ‘He

Each of the superfluid phases of 3He breaks a unique group of continuous symmetries. What is more, with their respective relative symmetry,

each embodies the only realization in nature of a

characteristic hydrodynamic response. There-

fore, from the hydrodynamic point of view, they

behave as differently from each other as, say,

superfluids and crystals. Helium is indeed more than the fixation of a small group of physicists

who have forgotten the rest of the periodic table.

3.1. -‘He-B

The ‘He-B phase, or rather the Balian-Wer-

thamer state, has four additional hydrodynamic variables. They are the spin-orbit rotation angle

dn, = d& - R,dOS, and cp, the phase. Rim denotes the order parameter, a spin-orbit rotation matrix. So the B-phase is such that its change under an infinitesimal orbital rotation d@ can be

undone by one in spin space, if d@ = R,, de;. A deviation from this invariant combination, lead-

ing to a non-vanishing dn,, is the broken sym-

metry of the system. The first equation of motion

is given by eq. (4), and the second by combining

eqs. (3) and (5) according to the definition of dni:

dnJdt - 0, + Rj,w, = 0. c-9

The B-phase is therefore a biaxially antiferromagnetic, biaxially nematic super&id, with an

inherent confusion between the vorticity and spin angular velocity, rotated by Ri,. This is the reason we expect the effects, mentioned in the

introduction, of mechanically excited spin waves

and NMR or of magnetically induced shear in-

stabilities.

3.2. -‘He-A

The A-phase, or rather the Anderson-Brink-

man-Morel state, contains five additional hydrodynamic variables. They are the two transverse rotations in spin space dd = de” X d and the GOS-variable d0 = de” - 2 dq, where d and ,? are two preferred directions, in spin and orbital space, respectively. The equations of motion appear in the following combinations:

M. Liu I Broken relafive symmetry of supe@tid ‘He I h7.7

dcildt = 0 x d (9

dtVdt = L? + (2mplii)l. (1W

Employing [S] dv, = -(h/2m)< V de;, we can extract the superfluid Euler equation

d, + V(/_l + V, * 0,) + (fi!2m)~, Vfli

= U” x (V x v,) (11)

Spin rotation around & in the absence of a

strong magnetic field, is not a broken symmetry. We conclude that the A-phase is a uniaxially antiferromagnetic, biaxially nematic superfluid without the ability to discern the chemical potential from the vorticity along 2. This is the

origin of the “gauge wheel” effect described in the introduction.

Similar effects, as pointed out by Ho and Mermin 1211, exist in a current-carrying state of

superfluid 4He. This appears quite surprising at

first, considering the simple gauge invariance broken in 4He-II on one hand and the causal

relation between GOS and gauge wheels on the

other. But it is easily explained. A state with a

constant 6, breaks the relative translation-gauge symmetry. i.e. its change under the gauge trans-

formation dq can be undone by the translation of (m/h)u, * du. The system therefore behaves as

a superfluid smectic, but will fail to distinguish /.L for v,. A shear flow, representing a non-uniform

field simultaneously of R and u,, will therefore

wind up the phase and drive a supercurrent in

both ‘He-A and the current carrying state of 4He. An interesting question in this context is the difference between the current carrying 4He and

the Fulde-Ferrell state [22] that also breaks relative translation-gauge symmetry, albeit in currentless true equilibrium.

3.3. ‘He-A,

The Al-phase again has two preferred directions, h^ and i, in spin and orbital space, respectively. Each breaks two rotational symmetries,

d& = dtP x h^, and di = dtP X 2. In addition. there

is the variable of relative SOGS, d4 =

dp - ? . de” - 6 . de”, that is a linear com-

bination of three symmetry transformations. As

a result. the system is a biaxially antifkrromag- netic, biaxially nematic superjluid, whose response to the three physical fields of chemical

potential. vorticity along 8, and spin angular velocity along & are. within certain limits. iden-

tical. Apart from the inherent confusions of the

A- and B-phases, it is the interchangeability be-

tween the chemical potential and the high frequency magnetic field that is the fingerprint of the Al-phase. In other words, the simple model

of subsection 2.4 is in fact a realistic one and displays characteristic Al-behavior.

In what follows, I shall set up the equations of a three-fluid hydrodynamics. by emulating Lan- dau’s original concept of superfluidity, to des- cribe both the A,- and the A-phase in a strong

magnetic field. The resulting equations. linearized and limited to the lowest order in 4.

are very transparent as to their physical content and a great help for an intuitive understanding of

the two phases. We have one normal lluid and two superfluids. The superfluids are charac-

terized, respectively, by their density tensors. pr

and ~1, their spin densities (h/2rn‘)pI and

-(h/2m)pL, and their vanishing entropies (per

unit mass), al = (~1 = 0. The normal fluid then obviously has to carry the density pn =

p-(p, +p~), the spin S,=S-(h!2m)(pt - pi). and the entropy .7,=s-(prcr* +pLcrJ)=s. These facts are expressed in the three transport equations:

b+V@,V,+PtUt +Plvl)=o, (12)

s + V(sv,) = 0 , (13)

S+V[S,V,+(h/2m)~(p~v~ -PiuJ)]=h. (14)

where A is the dipole torque. In addition, there are the three Euler equations:

1624 M. Liu I Broken relaiive symmetry of superj7uid ‘He

6, +pj’Vpr = -(UT - Ui)/T, (16)

;, +pj’vpJ = _(?JJ - Ut)/T. (17)

The partial pressures are again determined by

the fluids’ “load”:

Sp, = p,,S/a~ + S,So + SST,

SPT = Pr[& + W~WI 2

spl = PJ[SI* - (h/2m)Sw] .

The sum of the three gives the total measured

pressure. r denotes the dipole relaxation time. In the static limit, to? = vi, and a current-carrying

state transports both spin and mass, with a tem-

perature and field dependent ratio of (h/2m) x

(p T - p 1 )l(P t + P J), approximately (Wm > X

[ 1 + S(T, - T)/3(T, - T2)]-l in the Ginzburg- Landau regime. (Tl,2 refers to the A1,2 transition

temperature.) In the Al-phase, pi = A = 7-l = 0. Discarding

eq. (17) we are left with a two-fluid system. Note

that (i) eq. (16) is then identical to eq. (6) and (ii) with u1 entering both eqs. (12) and (14) it is

simultaneously a spin and mass current. The

Goldstone mode of the system, as we already

know, is a pair of spin-temperature waves. But now it is easy to see that the propagation arises

from the fluctuation in the counter flow, ~1, - u,,

between the ferromagnetic superfluid and the

(less-than) paramagnetic normal fluid, with the total density kept constant. Because the regions of

increased normal density are higher in tem-

perature but lower in spin content than those of

concentrated superfluid, both spin and temperature participate in this mode, leading to the velocity (c: + cf)‘“, where c,, c2, and c4 below denote the usual velocity expressions for spin wave, second, and fourth sound, respectively. All these have been clearly verified by Corruccini and

Osheroff [lo]. Putting ti, = 0 in eq. (16) yields the magnetic

fountain effect. It vanishes instantly, once the system undergoes the AT-transition. This is due

to two reasons. First, the dipole relaxation is

very effective in eliminating any excess Aw. Secondly, irrespective of how large pi is, eq. (17) is now hydrodynamically relevant. With tj, = tic = 0, the only solution satisfying both eqs. (16)

and (17) is Sp = SW = 0.

The influence of the transition on dynamics is

more involved, but of interest even beyond the

context of 3He. The AT-transition is the only

known case of a system going from a state of a

broken relative symmetry to one with two independently broken symmetries; in other words,

instead of breaking two symmetries in suc- cession, as is usually the case, “He chooses to

break one linear combination first and then the other to complete the breaking of both symmetries. And the problem here is how the cor-

responding Goldstone modes accommodate to

the altered “path” of symmetry breaking. Generally speaking, the second transition can

take place in two distinct fashions. If the two

constituent velocities are rather different, the

transition can be considered as characterized by the onset of the mode with the slower of the two

velocities. If they are comparable, it is marked by the same hybrid mode that would have marked the transition, had the transition been the

first one. In other words, the constituency of the

Goldstone mode is in this case independent of

the order of symmetry breaking and reflects only

the linear combination of that relative symmetry, whose breaking defines the transition. In 3He, with c4% c,% c2, only the first case is relevant,

and the A2-transition can be taken as marked by the onset of spin wave in a restricted geometry and by that of second sound generally [20].

A number of authors have studied the A- phase hydrodynamics close to the AZ-transition. Gongadze, Gurgenishvili, and Kharadze 1231 were the first to have calculated (i) the hybridiza- tion between spin wave and fourth sound and (ii)

the mixing of spin and entropy in spin wave and second sound. Saslow and Hu [24] and Takagi [25] have worked out the general, non-linear hydrodynamics with subsequent detailed discussions.

M. Liu 1 Broken relative symmetry of superfluid ‘He

Brand and Pleiner [26] predicted a magnetic spin or orbital space; hence, V x u, = 0. It seems

fountain effect. similar to the one in the Al-phase therefore quite reasonable to expect rotational

by (i) postulating terms equivalent to the entropy responses that are similar, even identical, to

densities. CT? and ‘TV, of the superfluid spin popu- those of “He-II. Let us, however. consider the

lations, and (ii) neglecting the dipole torque and system’s response in rotating equilibrium.

the dipole relaxation time. Finally, in a paper by Assuming o, = 0, the kinetic energy is given by

the present author, some aspects are studied from ni dL,. An expansion of SL, and SS, in 60, and

the point of view presented here [20]. 60, yields

4. Rotating equilibrium

The Andronishkavilli experiment is of crucial

importance to our understanding of 4He-II. It

demonstrates the superfluid’s unusual static response to rotations. This response follows unam-

biguously from the irrotationality of u, and the

Galilean invariance of the system. As a direct

consequence of the latter [19], we have

g = ~0, + j, W)

that, together with a second thermodynamic

relation, u, = j,/p, + v,, determines the “kinetic”

part of the energy. The angular momentum density is given, just as in the normal phase, by

L=rXg. (19)

If y5 = 0, we have

(20)

where 0, = p(r’Si, - rlr,) is the moment of inertia density of any normal liquid. The situation

changes in the superfluid phases of 3He. Here, rotating equilibrium yields independent information from the translating one.

4.1. ‘He-B

‘He-B is a “straight” superfluid. Its gauge invariance is broken independently, and a phase change cannot be undone by any rotations, in

In words, because of the order parameter’s symmetry, a cross thermodynamic derivative -pF becomes possible and couples spin and orbital rotations. Above T,, (Y, = ps = 0; approaching

zero temperature, with pn vanishing, the thermodynamic stability requires (Y, 2 /3$y’!. In fact, taking R,,6L, + SS,, = 0 at T = 0 yields PF = 1 and

ff, = XIY2.

Note that eq. (18), as a result of translating

equilibrium, remains rigorously valid. forcing

one to abandon eq. (19), and thus leading to a

natural definition of an intrinsic angular

momentum. Eq. (19) can of course be preserved

by amending eq. (1X) with terms -VW,, V,R,. But the resulting form evidently goes beyond the

local equilibrium approximation. (The reader

who is unsure of this statement should refer to

subsection 2.1, the results of which are heavily relied upon in this section.) A more subtle point

is the definition of w, = WC~S,. In the rotating

equilibrium L is held constant when taking the derivatives; in the translating equilibrium g is held constant. The difference between the two

definitions, however. can be shown to persist only for microscopic times, as long as f2 is not given by ;V x v,.

Now we are ready to discuss a few consequences of eqs. (21) and (22). In equilibrium,

a = o. and an Andronishkavilli experiment will measure the total moment of inertia,

J dvol[(p,lp)& + asail - /i$f_~Jy’)R,,]. while an appropriately located coil will sense the mag-

1626 M. Liu 1 Broken relative symmetry of superfluid .‘He

netization variable. Similar results are reported by Com- bescot and Dombre [28]. On the macroscopic

side, Nagai [32] and Combescot [29] examined the consequences of a higher order expansion,

essentially in the hydrodynamic variables, while Brand and Pleiner [33], and Saslow and Hu [34],

undertook a genuine next order hydrodynamic

expansion. A different line of reasoning under-

lies the works by Hall [35], and by Miyake,

Takagi and Usui [36]. They have studied the hydrodynamic consequences by explicitly taking

into account the presumed existence of the intrinsic angular momentum. Finally, Ho and

Mermin [37], Miyake and Usui [38], and Ashida [39] have all, from different points of view,

stressed the importance of rotating equilibrium. This is what we are now going to examine. The off-diagonal terms in the change of the angular

momentum density, t[a,~6p + a2cST], lead, just as in the case of the B-phase, to modifications of

familiar thermodynamic relations:

ma = XV-L + (Sk - &&)~,lr1 (23)

This is the “modified Barnett effect” predicted

by Combescot [27]. He and Dombre [28] have also microscopically calculated the B-phase

dynamics at zero temperature and studied the phenomenological approach with a higher order

expansion [29], albeit in the hydrodynamic vari-

ables. Turning now to dynamics, we take L?, w, etc.

as spatially and temporally varying functions but, in accordance with the concept of local equili-

brium, neglect possible higher order terms -VLJ,

VW . . . in the energy. Inserting eq. (22) into (8) leads to the important modification noted by

Combescot :

7j; - Ri,y*SSJX + (1 - ps)tii = 0 . (24)

Note that its stationary solution requires only that fli and w, are constant, not necessarily equal. This is valid also for a time span small

compared to the spin-orbit relaxation time, while eq. (23) gives the true static response.

4.2. ‘He-A

The A-phase intrinsic angular momentum has,

from the very beginning [l, 2,4], been a subject in search of consensus, leading to periodic flaring up of controversies. Dividing the recent papers,

according to their approach, into the microscopic and macroscopic categories, we have altogether ten of the former and twelve of the latter. Dis-

criminating Cooper pairs from Bose-condensed diatomic molecules, Mermin and Muzikar [30]

have found, in a weak coupling theory, that the momentum density contains a term -VP which can be interpreted as an intrinsic angular momentum density. Nagai [31] investigated the complete A-phase dynamics and found, in the zero temperature limit, a vanishing contribution of the vorticity to the time evolution of the phase

Sp= K(S/.L +U,l$fIi),

SS = C(ST+ U*@f2i),

(25)

(26)

where the thermal expansion coefficient is neglected. Above T,, a, = u2 = 0; approaching

zero temperature and assuming SLi = k’i((h/2m)Sp, we have ~1 = h/2m. Note Vip =

u,~nV,j, V;s = u2cflV& in equilibrium [31,32].

Now we turn to dynamics. Inserting eq. (25) in eq. (11) we obtain

dv,ldt + v,~VV,,~ + VP/K

+ (W2m - al)f?iVQ - U,niV~i = dissterms.

Essentially this equation was derived and discussed in detail by Hall, and Nagai was the first to derive its linearized version.

Similar thermodynamic discussions, yielding

modifications that grow slowly below T,, may seem somewhat academic in the Al-phase. However, with its peculiar symmetry, it does have the highest number of cross derivatives.

M. Liu I Broken relative wmmetry of superfluid ‘He

As a concluding remark, it should be stressed that the actual value of the intrinsic angular

momentum seems to be irrelevant in our present context. It is rather the angular momentum’s response that we have studied and found to be

important. The difference between these two has been examined by Volovik and Mineev in their

recent preprint [40].

As a side line, the basic concepts underlying

the hydrodynamic theory, especially local equilibrium and gradient expansion, are discussed with some care. It is then shown that the recent microscopic results are quite consistent with the

hydrodynamic expressions containing only the

lowest order, local equilibrium terms

References 5. Conclusions

The central subject of this paper is the coexis-

tence of different broken symmetries. We started with a discussion of four basic systems, each

breaking one type of symmetry. The common

feature of these systems is the emergence of a new flux. driven by an inhomogeneity in the respective relevant field: the spin angular velo-

city drives a spin current in antiferromagnets, the

vorticity drives an angular momentum flux in

nematics, the chemical potential drives a mass current in superfluids, and the velocity drives a

momentum flux in crystals. We then concluded that two types of coexisting broken symmetries provide the system with two additional currents

and two relevant fields.

[I] A.J. Leggett, Rev. Mod. Phys. 47 (1975) 331. [2] P.W. Anderson and W.F. Brinkman, in: The Physics of

Liquid and Solid Helium, K.H. Bennemann and J.B. Ketterson, eds. (Wiley, New York, 1977).

[3] P. Wolfle, Rep. Progr. Phys. 42 (1979) 269. [4] W.F. Brinkman and M.C. Cross, in: Progress in Low

Temperature Physics, vol. VIIA. D.F. Brewer, ed. (North Holland, Amsterdam, 1978).

[S] P.G. de Gennes. The Physics of Liquid Crystals (Clarendon, Oxford, 1974).

[6] M. Liu and M.C. Cross, Phys. Rev. Lett. 31 (1978) 250. [7] R. Graham, Phys. Rev. Lett. 33 (1074) 1413: M. Liu.

Phys. Rev. B 13 (1976) 4174. [S] M. Liu and M.C. Cross. Phys. Rev. Lett. 43 (1979) 296. [9] M. Liu, Phys. Rev. Lett. 43 (1979) 1740.

[lo] L.R. Corruccini and D.D. Osheroff. Phys. Rev. Lett. 35 (19X0) 2029.

[ 1 l] L.D. Landau and E.M. Lifshitz, Theorv of Elasticity, (Pergamon. Oxford, 1970).

A certain twist is added to this scheme by the

occurrence of broken relative symmetries, which is the breaking of a linear combination of two or

more continuous symmetries. A system of

broken relative symmetry behaves as if it broke

all the constituent symmetries but lacked the capability to distinguish between the various

relevant fields. In addition, a flux emerges that transports all the corresponding conserved

quantities at the same time. As it turns out, all three phases of superfluid ‘He break at least one relative symmetry. Accordingly, the B-phase is

confused between the vorticity and the magnetic field, the A-phase fails to separate the chemical potential from the vorticity along 2, and the Al-phase cannot discern the longitudinal spin

angular velocity from the above two fields of the A-phase.

1121

1131

[I41

I151

P.C. Martin, P. Parodi. and P.S. Pershan. Phys. Rev. A6 (1972) 2401; F. Jahnig and H. Schmidt. Ann. Phys. (NY) 71 (1972) 129. One might think of isotropic ferromagnets and antiferromagnets as an exception to thts rule. Despite the fact that both share the same broken continuous symmetries. their collective spectrum dithers considerably. It is, of course, the “pathological” behavior of ferromagnets to have the conserved magnetization acting simultaneously as the order parameter that spoils the analogy. In superfluid helium or crystals this pathology would cor- respond to the degeneracy of phase and number density or of displacement vector and momentum density. respectively. Cf. B.I. Halperin and P.C. Hohenberg. Phys. Rev. IX8 (1969) X98. In cases lacking true long-range order, it reflects the non-vanishing static response to an inhomogeneous transformation. Cf. H.J. Mikeska and H. Schmidt. J. Low Temp. Phys. 2 (1970) 371. This is a sweeping statement that. although correct in most circumstances, can and does become invalid occasionally such as at the critical point. Cf. I.M. Khalatnikov. Theory of Superfluidity (Benjamin, New York, 1965). More recent applications are: M. Liu, Phys.

1627

1628 M. Liu I Broken relative symmetry of superfluid ‘He

Rev. Lett. 35 (1975) 1577, and Phys. Rev. A 19 (1979) [24] W.M. Saslow and C.-R. Hu, Phys. Rev. B 23 (1981) 2090. 4523.

[16] M. Liu, Phys. Rev. A 24 (1981) 2720.

[17] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Ad- dison-Wesley, New York, 1959).

[18] We are accustomed to the idea that the density p propagates, while the entropy s/p diffuses. This does not appear to match the fact that e is a universal means of transport. However, the discrimination between p and s arises from a biased viewpoint and in fact derives from a numerical accident. If sap/&* paplap were true instead of the reverse case, we would probably connect s with propagation and p/s with diffusion.

[19] A.F. Andreev and I.M. Lifshitz, Zh. Eksp. Teor. Fiz. 56 (1969) 2057 [Sov. Phys. JETP 29 (1969) 11071: M. Liu, Phys. Rev. B 18 (1978) 1165.

[25] H. Takagi, Progr. Theor. Phys. 65 (1981) 1145. [26] H. Brand and H. Pleiner, J. Phys. C 14 (1981) 97. [27] R. Combescot, Phys. Lett. 78A (1980) 85. [28] R. Combescot and T. Dombre, Phys. Lett. 76A (1980)

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print.

[20] M. Liu, Z. Phys. B 40 (1980) 175. 1211 T.L. Ho and N.D. Mermin, Phys. Rev. Lett. 44 (1980)

330.

[33] H. Brand and H. Pleiner, Phys. Rev. B 23 (1981) 155. [34] W.M. Saslow and C-R. Hu, preprint. [35] H.E. Hall, preprint. [36] K. Miyake, H. Takagi and T. Usui, Progr. Theor. Phys.

65 (1981) 1115.

[22] P. Fulde and R.A. Ferrell, Phys. Rev. A 135 (1964) 550. [23] A.D. Gongadze, G.E. Gurgenishvili and G.A.

Kharadze, Zh. Eksp. Teor. Fiz. 75 (1978) 1504 [Sov. Phys. JETP 48(4) (1978) 7591; G.E. Gurgenishvili and G.A. Kharadze Zh. Eksp. Teor. Fiz. 31 (1980) 593 [JETP Lett. 31 (1980) 5571.

[37] T.L. Ho and N.D. Mermin, Phys. Rev. B 21 (1980) 5190. [38] K. Miyake and T. Usui, Progr. Theor. Phys. 64 (1980)

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