Why A phase?

The very important question which arose soon after the phase diagram was discovered experimentally, is “Why should the A phase be there?”. To see this problem, remember the Ginzburg–Landau theory discussed in the previous section. For the pairing with givenℓ, it was shown that the free energy is minimized by the choice which minimizes the anisotropy of |d( ˆn)|² over the Fermi surface. We can define this anisotropy by

K ≡ |d|⁴

(|d|²)2, (3.10)

and all we have to do is to minimize K.

Now we can estimateK by using the results of the gap function of each state, and see:

3Actually, there was an earlier discussion of this phase by Yu. Vdovin [3], but this work was not widely known outside the former Soviet Union.

A. J. Leggett LEC. 3. SUPERFLUID³HE (CONTINUED)

For the ABM state, |d( ˆn)|² ∝sin²θ ⇒KABM = _(2/3)^8/152 = 6/5.

For the BW state, |d( ˆn)|² = const. ⇒KBW = 1.

Thus, in the generalized BCS theory, we always haveFBW < FABM!

This was the major problem when superfluid phases of³He were discovered experimen-tally, and some approaches were developed to examine this problem. The first approach is to generalize Ginzburg–Landau approach, and the second one is to consider spin fluc-tuation feedback. In the next two subsections, I will introduce these two approaches.

3.3.1 Generalized Ginzburg–Landau approach

In this approach, our discussion is only based on the symmetry of the states, and there is no particular assumption on energetics. Since d( ˆn) is a vector in spin space, and the orbital dependence is assumed to be p-wave, we can always write the quantities dα as

dα( ˆn)≡∑

i

diαnˆ_i, (3.11)

where i is a spin subscript, and α is an orbital one. This means that all p-wave triplet states are completely parameterized by 9 complex quantitiesdiα, and the order parameter is defined in the 18-dimensional space.

Here, let us follow the original argument by Ginzburg and Landau, and assume the expansion:

F(T :{diα}) =α0(T −Tc)·O(|d|²) +β(T)·O(|d|⁴) +..., (3.12) where α0 >0 is a constant. Then, we consider some basic symmetries: invariance under global gauge transformations, and under rotations of the spin and orbital coordinate system separately. These symmetry conditions constrain the term of O(|d|²) to have the unique form

O(|d|²)∝∑

iα

|diα|², (3.13)

which is ∫ _dΩ

4π|d( ˆn)|², just as in the BCS theory.

For the O(|d|⁴) term, on the other hand, there arefive terms which are invariant under the above symmetries. For example,

I₁ =

∑

iα

d²_iα

2

, (3.14)

I2 =∑

αβij

d^∗_βid^∗_βjdαidαj, etc. (3.15) Here, note thatI₁ is different fromI₄ ≡(∑

iα|d_iα|²)² = 1. We can write down any fourth-order invariants by the linear combinations of these five invariants⁴. Therefore in general,

A. J. Leggett LEC. 3. SUPERFLUID³HE (CONTINUED)

setting some coefficients βs, we obtain O(|d|⁴) =

5

∑

s=1

β_sI_s ≡K₄. (3.16)

Is’s are fourth-order invariants characteristic of particular kinds of states (e.g., the ABM state, the BW state ...), but βs’s are parameters which depend on energetic assumptions in general. Therefore we cannot determine βs a priori unlike the simple BCS case.

Next, we should tackle the problem of findingdiα, which minimize the fourth-order free energy K4 for given parameters βs and normalization conditions:

∑

iα

|diα|² ≡ |d|² = 1. (3.17) The problem is to find all possible states (all possible forms of diα) which can be minima of free energy (i.e., minima of K4) for some choice of theβs under the above constraints.

This problem is solvable in principle, but is quite messy and nearly unsolvable in reality without constraints. Therefore we restrict possible states to unitary states. Here, unitary states mean that for every ˆn, Sz = 0 is satisfied in some set of spin axes.

If we assume this, the problem becomes simple, and it is found that only four states can be the extrema of free energy (for more details, see Ref. [5]):

1. The BW (“isotropic”) state: dαi = 1

√3δαi

This state is already introduced. In the BW state, we have pairing in all possible directions.

2. The 2D (“planar”) state: dαi = 1

√2δαi(1−δiz)

This state is essentially very similar to the BW state because if the z-component is removed from the BW state, it becomes the planar state. This state is an ESP state, but is different from the ABM state.

3. The ABM (“axial”) state: dyx=−idzx= 1

√2, all other dαi = 0 This state is already introduced.

4. The 1D (“polar”) state: dzz = 1, all otherdαi = 0 This state is also an ESP state.

Remarkable Theorem

For the BCS values of β^′s, the BW state turns out to be most stable. If the non-BCS contributions to β are taken into account, however, other states may be more stable. In fact, the polar state can be more stable than the BW state only if the non-BCS contribu-tions to β^′s are comparable to BCS ones. For the ABM state, its energy become smaller

A. J. Leggett LEC. 3. SUPERFLUID³HE (CONTINUED)

than that of the BW state for relatively small non-BCS contributions. Furthermore, we can show that the 2D (planar) state can never be the absolute minimum of free energy for any choice of β^′s. Therefore, in general, the BW or the ABM states are most likely to appear.

It is important to note that in the above analysis, both orbital subscriptsα’s and spin subscriptsi’s always occur in pairs. Thus,I_s^′sare invariant under spin and orbital rotation separately, and the BW state, the ABM state, etc. representclassesof states transforming into one another under these rotations.

3.3.2 Spin fluctuation feedback

In this subsection, we view the second approach which is much more based on physical mechanism. This approach is proposed by Anderson and Brinkman in 1973 [6].

The basic physical idea of Anderson–Brinkman theory is shown in Fig. 3.3. In a stan-dard superconductor, which is described by the BCS theory, the mechanism of the forma-tion of Cooper pairs is an exchange of virtual bosons, or phonons between the electrons. In the case of ³He, quasiparticles also exchange bosonic degree of freedom called “spin fluc-tuation”, where additional quasiparticle and quasihole pairs are formed and the medium becomes virtually polarized by the strong inter-atomic interaction. Thus, the mechanism for the superfluidity sounds rather similar to the BCS one, but there is a critical difference.

Unlike the superconductor, in the case of³He, the medium being polarized is precisely the medium in which Cooper pairs are formed, and the superfluid phase transition modifies the pairing interaction between quasiparticles. Therefore, the original force of the super-fluidity, the spin fluctuation, is in turn modified in the presence of the supersuper-fluidity, and in general, we must include this “spin-fluctuation feedback”. The amount of this feedback effect depends on a particular kind of superfluid (the ABM state, the BW state, ...), and it is, in fact, significant in discussing the ABM and the BW states.

The spin-fluctuation-induced interaction (See Sec. 2.3) is given by

Vˆeff(qω)≈ −(F₀^a)²χsp(qω)σ⁽¹⁾·σ⁽²⁾, (3.18) whereσ⁽¹⁾ andσ⁽²⁾ are spin operators for each atom of the Cooper pairs. This interaction is attractive in the spin-triplet state, and repulsive in the singlet state. The point is that the spin susceptibility χsp(qω) is modified by pairing. To obtain quantitative results, we need a microscopic complicated calculation [6], but we can easily obtain qualitative understanding by assuming

δχsp(qω)∝δχ, (3.19)

whereχis a static spin susceptibility. Crudely speaking, we assume that the modification of the spin susceptibility is similar to the static part. With this assumption, we can immediately see that the BW state is disfavored. In fact, in the BW phase, the

spin-A. J. Leggett LEC. 3. SUPERFLUID³HE (CONTINUED)

Lattice vibration,

insensitive to onset of electron pairing

Superconductor:

Liquid

He:

Spin fluctuation of ³He system, sensitive to onset of pairing

=

Fig. 3.3. Schematic illustration of the Anderson–Brinkman theory.

In the ABM phase, for fixed d, δχis actuallyanisotropic. In this case, we can write down the modification δχ and potential as

δχ_ij ≈ −f(T)d_id_j, (3.20)

∆hVˆeffi ≈ −δχijhσ⁽¹⁾_i σ_j⁽²⁾i ≈+f(T)didjhσ_i⁽¹⁾σ⁽²⁾_j i. (3.21) If we take the axis of d in the z-direction, we find ∆hV_effi ∝ d²_zhσ⁽¹⁾z σz⁽²⁾i. In the ABM state, all Cooper pairs are in a state withS = 1, Sz = 0, so that the spins of the pairs are in the z-direction are antiparallel and hσ⁽¹⁾z σz⁽²⁾i is strongly negative. Hence, in the ABM state, the spin-fluctuation attraction is increased in the ABM state over the normal-state value. If the spin-fluctuation feedback effect is strong enough, the ABM state may become even more stable than the BW state.