Cooper L.N., Feldman D. (Eds.) BCS: 50 Years

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September 17, 2010 9:45 World Scientiﬁc Review Volume - 9.75in x 6.5in ch18

The Superﬂuid Phases of Liquid

He: BCS Theory 481

state is bound as in Cooper’s original calculation (for details see e.g. Ref. 2,

Sec. 5.4). In the l 6= 0 case the behavior both of F (r) and of the energy

gap function ∆(k) is anisotropic, with ∆(k) having roughly the form of a

combination of spherical harmonics Y

(

k) (and, as in the s-wave case, little

dependence on |k|). Thus, while the spin properties of the system described

by Eq. (3) should be isotropic as for the BCS case, orbital properties such as

the (nonzero-T ) superﬂuid density may display an experimentally measur-

able anisotropy (cf. Ref. 3). This would already be a signiﬁcant diﬀerence

from the BCS case.

However, the case of spin triplet (odd-l) pairing is even more interesting.

Let us specialize for simplicity of presentation to the p-wave case (which as

we shall see is generally believed to be that relevant to all three new phases

of superﬂuid

He). We then have three possible Zeeman spin substates and

three orbital ones, giving a nine-dimensional Hilbert space. Quite generally,

we can resolve the pair wave function in terms of its amplitudes in the spin

Zeeman substates: again using an obvious notation,

F (r : σσ

) = F

↑↑

(r)|↑↑i + F

↑↓

(r) ·

√

(|↑↓ + ↓↑i + F

↓↓

(r)|↓↓i (5)

Since we are interested in the present context in the angular dependence

of the relative wave function, it is convenient to take the Fourier transform

F (k) and sum it over the energy variable |k| (recall that in the BCS limit

 E

, which should be a fairly good approximation for superﬂuid

where T

∼ 10

−3

, the sum goes only over a range  k

). As a function

of angle

n on the Fermi surface the resulting (complex) expression is closely

proportional to the (complex) energy gap ∆(

n), so it is conventional to

discuss the latter, writing it in a form similar to (5):

∆(k : σσ

) = ∆

↑↑

(k)|↑↑i + ∆

↑↓

(k)

√

|↑↓ + ↓↑i + ∆

↓↓

(k)|↓↓i (6)

One more convenient piece of notation and we are through. If we for the

moment put aside the A

phase, it turns out that all the p-wave Cooper-

paired phases seriously considered as candidates for the experimentally ob-

served A and B phases of superﬂuid

He have the following (“unitary”)

property: for each direction of

k on the Fermi surface separately, we can

ﬁnd a choice of spin axes (call it choice (a)) such that of the above three

amplitudes, only ∆

↑↓

(

k) is nonzero. (We emphasize that the choice need not

be the same for all values of

k.) Let us deﬁne, for any given direction

k, a

unit vector

d(k) as the z-axis in this basis; then a compact description of

the symmetry of any given state, which is entirely equivalent to Eq. (6), is

September 17, 2010 9:45 World Scientiﬁc Review Volume - 9.75in x 6.5in ch18

482 A. J. Leggett

given by simply specifying the quantity d(k) ≡ |∆

↑↓

(

k)|

k) as a function

of direction

k on the Fermi surface. The precise equivalence is given by the

formulae (applicable for any k-independent choice of spin axes)

∆

↑↑

(k) =



(k) + id

(k)



, ∆

↑↓

(k) = d

(k),

∆

↓↓

(k) =



(k) − id

(k)



(7)

The quantity ∆

(k) ≡ |d(k)| =



|∆

↑↓

(k)|

+ |∆

↑↑

(k)|



1/2

is the gap mag-

nitude for direction

k on the Fermi surface, and plays the same role as

the “energy gap” in the original BCS theory, i.e. quasiparticle energies are

(

−µ)

+ |∆

(k)|

, etc. Note that for a unitary state it is also

always possible, for any given

k, to ﬁnd a choice of spin axes (diﬀerent from

the above one; call it choice (b)) such that the z-component vanishes; then

(as indeed more generally) the magnitude of the “up-up” gap is identical to

that of the “down-down” one though their phases may be diﬀerent.

The forms of d(

n) appropriate to the experimentally observed A and

B phases of superﬂuid

He have been inferred from various experimental

anisotropies, in particular that of the NMR data, and are also consistent

with theoretical predictions (see below); they are believed to be as follows.

The A phase is identiﬁed with the “ABM” phase (the one originally studied

by Anderson and Morel and later by Anderson and Brinkman); this is deﬁned

by the statement that

n) = df(

n) (8)

where d is a ﬁxed vector and with a suitable choice of orbital axes the

function f(

n) is

n) =





1/2

∆

sin θ · exp iϕ (9)

where θ and ϕ are the standard azimuthal angles and ∆

is the RMS gap

over the Fermi surface. In other words, Eq. (8) tells us that the spin wave

function factors out and with an appropriate choice of spin axes is simply

the S = 1, S

= 0 Zeeman state



√

(↑↓ + ↓↑)



; while Eq. (9) tells us

that with an appropriate choice of orbital axes the pairs appear to have a

relative angular momentum L

= 1 around the z-axis. (On the question of

the physical meaning of this observation, see below.) In a general reference

frame the spin state is characterized by a single vector d and the orbital

behavior by a single vector l; our considerations up to now give no reason for

any particular relative orientation of d and l. An interesting point about the

September 17, 2010 9:45 World Scientiﬁc Review Volume - 9.75in x 6.5in ch18

The Superﬂuid Phases of Liquid

He: BCS Theory 483

ABM phase is that it is one of a class of so-called “equal-spin-pairing (ESP)”

states that have the property that we can ﬁnd a choice (b) of spin axes such

that no matter where we are on the Fermi surface, up spins are paired only

with up and down only with down, so that to a ﬁrst approximation one can

regard the spin populations as independent. It is then fairly obvious that

application of a magnetic ﬁeld along this axis will simply enlarge the up-spin

Fermi surface relative to the down-spin one but will not otherwise interfere

with the pairing process (at least at this level of discussion). Thus, assuming

that the system is able to adjust the spin axes so that the “up” direction is

indeed along the ﬁeld, the susceptibility of the ABM phase should not be

reduced from that of the normal phase; this indeed seems to be the case to

a good approximation for the experimental A phase.

The B phase is usually identiﬁed with the “BW” phase predicted by

Vdovin and by Balian and Werthamer. The form of d(

n) that correponds

to the state explicitly considered by these authors is

n) = ∆

n (10)

which when translated into the language of Zeeman components describes

a state with each of the spin components paired with the opposite orbital

component, so as to give L = 1, S = 1 but J = L + S = 0; that is, what in

the language of atomic physics would be denoted a

state. By the Wigner–

Eckhardt theorem such a state must be isotropic in all its properties, since

it possesses no characteristic vector. Moreover, since with any choice of the

spin axes one-third of the spins are paired “oppositely,” the susceptibility is

reduced from the normal-state value (actually by more than 1/3, because of

molecular-ﬁeld eﬀects); such a reduction is indeed seen in the experimental B

phase (and as a result the B phase is unstable with respect to the A phase at

T = 0 in ﬁelds above ∼0.55T). However, it turns out that the straightforward

identiﬁcation of the B phase with the simple

state misses a subtlety;

since the latter does not aﬀect the discussion of the next few paragraphs, we

postpone it for the moment.

At this point let us make a brief digression to discuss the A

phase. This

phase, which occurs only in high magnetic ﬁelds, is non-unitary and thus

description in terms of (a generalized) vector d(n), while possible, is not

particularly convenient; a more compact description is that the A

phase

is simply the A phase with only one spin population paired. This phase

has some unique properties, particularly as regards the “two-ﬂuid” descrip-

tion of its hydrodynamics (see e.g. Ref. 9), but there is no space to discuss

them here.

September 17, 2010 9:45 World Scientiﬁc Review Volume - 9.75in x 6.5in ch18

484 A. J. Leggett

Returning to the A and B phases, which take up the bulk of the phase

diagram, we note that if the above identiﬁcations are correct the very

existence of the A phase in zero magnetic ﬁeld poses a puzzle, since accord-

ing to the pre-1972 calculations, based on a straightforward generalization

of BCS theory, the ABM phase, like all equal-spin pairing phases, should be

unstable in zero ﬁeld with respect to the BW phase, which is indeed, within

this framework, the unique minimum of the free energy below T

. This puzzle

was brilliantly solved by Anderson and Brinkman,

who took up the earlier

idea that a substantial part of the attraction responsible for forming the pairs

might be due to the exchange of virtual spin ﬂuctuations, and then pointed

out a crucial diﬀerence with the situation in superconducting metals: in the

latter case, the attraction comes from the exchange of virtual phonons, which

are excitations of a diﬀerent system (the ions) from the electrons which un-

dergo the pairing; consequently, when pairs are formed the structure of the

virtual phonons is eﬀectively unchanged. By contrast, in the case of liquid

He the virtual spin ﬂuctuations whose exchange provides the attraction are

excitations of the very same system as is forming the pairs, and thus their

structure can be modiﬁed by the onset of pairing. Now one might expect

that the “strength” of the spin ﬂuctuation exchange might be somehow pro-

portional to the spin susceptibility of the liquid, and thus be reduced in the

BW state, which would thereby be disadvantaged with respect to an ESP

state. Although this argument as it stands is too crude, it turns out that

a microscopic calculation of this eﬀect

(which goes beyond standard BCS

theory, in fact invokes terms of higher order in T

than are usually kept

in that theory) not only conﬁrms the intuitive result that it favors any ESP

state over the BW state, but, remarkably, shows that the speciﬁc ABM state

is uniquely favored, and that its advantage is greatest in the high-T, high-

pressure part of the phase diagram where the A phase is actually found to

occur. This is probably the only respect in which a satisfactory microscopic

theory of the superﬂuid phases of

He needs to invoke ideas that go quali-

tatively beyond the original BCS scheme or straightforward generalizations

of it.

At the phenomenological level, on the other hand, there is a crucially im-

portant diﬀerence between superﬂuid

He and the superconductors so suc-

cessfully described by BCS theory: as already remarked, in the latter case the

internal structure of the Cooper-pair wave function is uniquely determined

by the energetics, so that there are no degrees of freedom associated with it.

In the case of superﬂuid

He, on the other hand, while the “shape” of the

pair wave function in both the A and the B phases is determined, as sketched

above, by the “gross” energies of the problem, i.e. the kinetic energy and

September 17, 2010 9:45 World Scientiﬁc Review Volume - 9.75in x 6.5in ch18

The Superﬂuid Phases of Liquid

He: BCS Theory 485

interatomic (hard-core and van der Waals) potential energy, the orientation

is not. As regards this orientataion, the gross energies in eﬀect constrain all

the pairs to behave in the same way, but do not determine in which way. The

situation may be regarded as analogous to that in an isotropic Heisenberg

ferromagnet, where the strong (exchange) forces constrain all the spins to

lie parallel, but say nothing about the common direction in which they lie

(see further below). In the case of

He this is most obvious in the A phase,

because while the characteristic spin vector d and the characteristic orbital

vector l must be the same for all pairs, nothing we have said so far in any

way constrains their orientations, either absolute or relative to one another.

However, it is also true in a more subtle way for the B phase, since nothing

(at least nothing said so far) prevents us starting with the familiar

state

as above, and then rotating the spin coordinates by some arbitrary amount

(i.e. by an arbitrary angle θ about an arbitary axis

ω) relative to the orbital

coordinates; since none of the energies considered so far couple the spin and

orbital coordinates, the state so obtained must be degenerate with the orig-

inal one. This property (which is shared by the A phase, in the sense that

a relative rotation of l and d cannot change the energy) is sometimes called

“spontaneously broken spin-orbit symmetry”.

Before investigating some of the consequences of this state of aﬀairs, let

us digress for a moment to explore the physical signiﬁcance of the orbital

vector l in the A phase. Formally, it is the direction such that, if it is cho-

sen as orbital z-axis and the azimuthal angle θ and polar angle ϕ deﬁned

in the conventional way, the relative wave function of the Cooper pairs has

the angular dependence sin θ · exp iϕ. If one were talking about a diatomic

molecule (at rest), then such an angular dependence of the relative wave

function would indicate clearly that the molecule has total angular momen-

tum L = L

= ~. Can we draw a similar conclusion for

He-A? That is, does

the system possess a total angular momentum along l, and if so what is its

magnitude? Curiously, while just about everyone agrees that the answer to

the ﬁrst question is yes, the answer to the second is after 35 years still not

universally agreed; the answers N~/2 ≡ L

, ∼(∆/E

and ∼(∆/E

)

all ﬁnd advocates in the literature. The question has proved very diﬃcult

to resolve experimentally; it is complicated by the fact that in real life the

direction of l is likely to vary in space (see next paragraph). For what it

is worth, the view of the present author is that the question may not even

be well deﬁned, so that even for an ideally uniform direction of l the total

angular momentum may depend on the way in which the superﬂuid phase

was reached. For further discussion, see Ref. 2, Appendix 7A.

September 17, 2010 9:45 World Scientiﬁc Review Volume - 9.75in x 6.5in ch18

486 A. J. Leggett

The fact that the gross energies do not determine the “orientation” of

the pair wave function means that this quantity may vary in space and/or

in time, and this gives rise to some fascinating phenomena, in particular,

the possibility of a variety of topological singularities which are a more

sophisticated analog of the simple Abrikosov vortices which are found in

(type-II) BCS superconductors. The study of the various possible singu-

larities in the A and B phases with the aid of the branch of pure mathe-

matics known as homotopy theory has been a major industry over the last

30 years, see e.g. Refs. 11 and 12. One very interesting output of these

studies has been the conclusion that in the A phase, given that close to

a wall of the sample cell the l-factor tends to orient itself normal to the

wall, any simply connected sample must contain at least two topological

singularities.

An even more fascinating consequence of the spontaneously broken spin-

orbit symmetry is a property which I will call “superﬂuid ampliﬁcation”.

To explain this, it is helpful to return to our ferromagnetic analogy. In a

ferromagnetic material such as Fe, the electron spins may be subject to a

weak magnetic ﬁeld. If we are above the Curie temperature, the spins behave

to a ﬁrst approximation independently; the single-spin Zeeman energy E

in competition with the much larger thermal energy k

T , and the resulting

polarization is of order E

T  1. Below the Curie temperature the

situation changes qualitatively: the “strong” (exchange) forces now constrain

all or most of the N spins to lie parallel, and as a result the energy diﬀerence

which competes with the thermal energy is no longer E

but rather N E

which is much larger than k

T . As a result the polarization becomes of

order unity in even very weak external ﬁelds.

The analog of this behavior in liquid

He is that the “strong” terms in

the Hamiltonian (kinetic energy, hard-core, van der Waals attraction, even

the “eﬀective” spin-spin interaction generated by exchange) are all invariant

under a relative rotation of the spin and orbital coordinates; thus in the

normal phase, where there is no strong correlation between the behavior

of diﬀerent pairs of atoms, these coordinates are eﬀectively uncorrelated.

However, there exists one term in the Hamiltonian which is not invariant

under the relative spin-orbit rotation, namely the electromagnetic interaction

of the nuclear dipole moments. Crudely speaking, this term has the eﬀect

that if two atoms have parallel spins they would prefer to orbit in a plane

parallel to the spins rather than one perpendicular (“end-over-end” rather

than “side-by-side,” just as would two small bar magnets). However, the

diﬀerence in energy between these two conﬁgurations (call it g

) is very

September 17, 2010 9:45 World Scientiﬁc Review Volume - 9.75in x 6.5in ch18

The Superﬂuid Phases of Liquid

He: BCS Theory 487

small indeed (∼ 10

−7

K even at the minimum interatomic distance), so in

the normal phase it cannot compete with the thermal energy and, as already

remarked, the spins and orbital coordinates of any given pair of atoms are

eﬀectively uncorrelated.

Now consider what happens in a superﬂuid (Cooper-paired) phase. The

“strong” terms in the Hamiltonian now conspire to force all the pairs to

behave in exactly the same way, but cannot determine in which way. Now

the tiny nuclear dipole-dipole interaction comes into its own: since all pairs

must now have the same relative orientation of their spins and orbital co-

ordinates, the diﬀerence between the “good” and “bad” orientations is now

not ∼g

but rather ∼N

, where N

∼ N is some intuitive measure of the

“number of Cooper pairs,” and this diﬀerence is now very much larger than

the thermal energy. Consequently, the relative orientation of the spin and

orbital coordinates will adjust itself so as to minimize the dipole energy. In

the A phase the result is that the characteristic spin vector d orients itself

parallel (or antiparallel) to the characteristic orbital vector l, while in the B

phase the spins of the pairs are rotated away from the “reference”

con-

ﬁguration by the angle cos

−1

(−1/4)

∼

104

◦

. Of course, these prescriptions

still leave part of the original degeneracy (e.g. in the A phase, the common

orientation of l and d) unbroken, and to break this residual degeneracy we

need to invoke other eﬀects such as those of the cell walls; see e.g. Ref. 13,

Sec. X.

These, then, are the equilibrium conﬁgurations in the A and B phases

respectively. Now, what happens if we somehow tilt the orientation away

from equilibrium? (As we shall see below, this can be done by applying an rf

magnetic ﬁeld, which acts on the spins but not on the orbital coordinates.)

It turns out that this results in a torque on the total spin of the system (and

an equal and opposite torque on the total orbital angular momentum), which

adds to the eﬀect of the external magnetic ﬁeld and gives rise to anomalous

NMR behavior. We can make this observation quantitative by exploiting

the fact that the “microscopic” frequencies of the system, such as the “gap

frequency” ∆/~, the quasiparticle collision rate, etc., are all high compared

to characteristic NMR frequencies; as a consequence, it is a good zeroth-order

approximation to describe the NMR behavior by a Born–Oppenheimer type

of approximation, in which we express the Hamiltonian as a function only of

the “slow” variables, which in this case are the total spin vector S of all the

atoms of the system (irrespective of whether they are paired or not) and the

orientation of the spins of the Cooper pairs. It is convenient to describe this

latter variable, schematically, by the (vector) angle of rotation θ relative to

September 17, 2010 9:45 World Scientiﬁc Review Volume - 9.75in x 6.5in ch18

488 A. J. Leggett

the equilibrium conﬁguration; it is essential to appreciate that the variables

θ and S are kinematically independent; indeed, they may be shown to satisfy

the commutation relations

, θ

] = iδ

(11)

The Hamiltonian can now be written in the form

H = H

(S, H) + H

(θ) (12)

where the ﬁrst term contains the Zeeman and exchange terms, while the

second expresses the dependence of the nuclear dipolar energy on θ. (It

is implicitly assumed, here, that the orbital coordinates of the pairs are

eﬀectively frozen over times of the order of the NMR period.) Then the

semiclassical equations of motion take the simple form

dS/dt = N

ext

− ∂H

/∂θ (13a)

dθ/dt = ∂H

/∂S (13b)

where N

ext

is the usual torque due to the external magnetic ﬁeld. These

equations can be solved for a variety of interesting situations; in particular,

it turns out that the linear NMR behavior can serve as a “ﬁngerprint” to

identify the structure of the order parameter. For an ESP phase with a single

d-vector, such as the ABM phase, a “Pythagorean” shift of the resonance

frequency is predicted:

(T ) = ω

+ ω

(T ) (ω

≡ γH) (14)

in agreement with what is observed experimentally in the A phase. (For

other types of ESP phase the resonance generally splits, a behavior for which

there is no evidence in the experiments.)

The situation with regard to the linear NMR behavior in the BW phase

is quite interesting. While the considerations given above about the dipole

energy determine only the angle of the spin-orbit rotation, not its axis, there

are both theoretical and experimental reasons to believe that there is a very

tiny energy which in the bulk liquid tends to orient this axis (call it

ω) along

the direction of the external magnetic ﬁeld H. If now we do a standard NMR

experiment in which the oscillating rf ﬁeld is perpendicular to H, solution

of Eq. (13b) shows that the eﬀect of this ﬁeld is, to lowest order, to change

only the direction of ˆω, not the angle of the rotation; thus the dipole energy

is essentially unchanged and the second term in Eq. (13a) is zero, so no shift

in the NMR frequency is expected. However, an oscillating rf ﬁeld applied

parallel to H rotates according to Eq. (13b) the spin coordinates around

September 17, 2010 9:45 World Scientiﬁc Review Volume - 9.75in x 6.5in ch18

The Superﬂuid Phases of Liquid

He: BCS Theory 489

H, changing the rotation angle and with it the dipole energy. We therefore

expect a nonzero-frequency longitudinal resonance — exactly what is seen in

He-B. Thus the experimentally observed NMR behavior strongly supports

the standard identiﬁcation of the A phase of superﬂuid

He with the ABM

phase and of the B phase with the BW phase.

The principle of ampliﬁcation of ultra-weak eﬀects, which as I have in-

dicated is most spectacularly exempliﬁed in the NMR behavior, may have

other applications. For example, it is predicted

that the A phase may

show a very weak but nevertheless “macroscopic” (i.e. extensive) electronic

ferromagnetism, and this eﬀect may have been observed.

Also, consider the

possibility of an electric dipole moment d

induced by the P-violating but

T-conserving part (the dominant part) of the neutral-current terms in the

weak interaction of particle physics. In any ordinary system, characterized

by a single vector, the angular momentum J, the Wigner–Eckhardt theorem

states that the only possibility is

= const · J (15)

which violates not only P but T ; hence it is not a candidate. However, the

Cooper pairs in the B phase of

He are unique in being characterized not by

a total angular momentum J but rather by the T-invariant pseudo-vector

hL × Si = const · ˆω. Hence symmetry permits a macroscopic electric dipole

moment of the form

= const · ˆω (16)

which violates P but not T .

Should future experiments detect such a dipole moment (which

calculations

suggest is likely to be very tiny but not necessarily hopelessly

so), it would constitute the ﬁrst ever example of a macroscopic violation of

parity due to the weak interaction.

In conclusion, while most of our understanding of the superﬂuid phases

of liquid

He is based ﬁrmly on a generalization of the original ideas of BCS,

these systems provide a richness of behavior that goes far beyond that seen in

the original application to superconductors. Indeed, with the possible excep-

tion of the subsequently discovered fractional quantum Hall eﬀect, we may

say that superﬂuid

He is the most sophisticated physical system of which we

can currently claim not just a qualitative but a quantitative understanding.

Acknowledgment

This work was partially supported by the National Science Foundation under

grant no. NSF DMR09-06921.

September 17, 2010 9:45 World Scientiﬁc Review Volume - 9.75in x 6.5in ch18

490 A. J. Leggett

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