Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

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24 Superﬂuid

He and the Cuprate Superconductors 1523

E{ın(p )} = E



p

(p)ın(p ) (24.3)









f (pp







)ın(p )ın(p







) ,

where the groundstate energy E

is a constant. The

coefficients  and f have as | p |→ p

the following

behavior

(p) = const.(| p | −p

) , (24.4)

f (pp







) = independent of | p |, | p



f ≡ (dn/d)

−1





+ F







(ˆp ·ˆp



) ,

where dn/d is the single-particle density of states

attheFermisurfaceandP

the Legendre polynomial.

(Clearly, the noninteracting system is the special case

 =(p

/m)(| p | − p

), f = 0).The above expression

for the energy,which is strictly diagonal in the quan-

tities ın(p ),is never exactly valid in an interacting

system, because of terms in the Hamiltonian which

lead to mutual scattering of quasiparticles between

states with different p, ; however, arguments based

on the Pauli principle show that it is self-consistent

to take the probability of scattering to tend to zero as

(| p | −p

)

, that is, faster than the excitation energy

of the quasiparticle in question (see e.g., [9]).

It is worth noting that in the above discussion it is

not assumed that the interactionterms in the Hamil-

tonian conserve the total momentum P (or spin S)

of the system. The question of whether or not there

is a one–one correspondence, as above, between the

groundstate and low-lying states of the interacting

and noninteracting systems is independent of that

consideration. If the total momentum is in fact con-

served,ase.g.in the case of liquid

He,then the quasi-

particle states (low-lying approximate energy eigen-

states) are exact eigenstates of total momentum with

eigenvalue

P =



pın(p ) (24.5)

(and one can make a similar statement for the total

spin). In the opposite case, the labeling is still valid,

but the total momentum operator is in general not

diagonal in the basis formed by the quantity ın(p ).

It should be strongly emphasized that for any

given real-lifesystemof fermionsthe validity of Lan-

dau Fermi-liquid theory is an assumption, which can

almost never be proved from first principles, but at

best can be substantiated by comparison of the pre-

dictions made on that hypothesis with experiment.It

should be stressed that some of these predictions are

nontrivial, in the sense that the number of predicted

relations between different experimental quantities

is larger than the number of a priori undetermined

parameters needed to characterize the fundamental

quantities (p)andf (pp







Is liquid

He in fact a Fermi liquid? If in the defini-

tions one were to treat “low-lying” as meaning “with

(single-quasiparticle) energies much less than 1K but

much greater than 3 mK”, then all the evidence is

that the answer is yes. Whenever the properties of

liquid

He in the normal phase well below 100 mK

have been measured, it has always proved possible

to fit the results into the framework of Fermi-liquid

theory. However, the very existence of the superﬂuid

phases shows that by a technically accurate defini-

tion liquid

He is ironically not a Fermi liquid. In-

deed, the excitation energies of the very low-lying

excited states relative to the groundstate are of the

generalized BCS form (see Sect. 24.4) and certainly

do not correspond to expressions (24.3) and (24.4).

This situation is usually summed up by saying that

liquid

He in the normal state behaves as a Fermi liq-

uid. It then follows that the phenomenon of Cooper

pairing can be discussed on the basis of this picture,

see Sect. 24.4.

As already remarked, the Landau Fermi-liquid

theory was originally developed with specific ref-

erence to liquid

He, which is both neutral and

translation-invariant. The generalization of the the-

ory to a charged but still translationallyinvariantsys-

tem was made soon thereafter by Silin [10]. In effect,

one simply generalizes the philosophy of the stan-

dard random phase approximation by using the orig-

inal Landau theory to calculate the “local” responses

andthen,when necessary,replacing theexternal elec-

tric field by the sum of the external field and that

produced self-consistently by the charge response of

the system itself. The generalization to the case of a

finite crystalline potential is a little more tricky and

1524 A.J.Leggett

is discussed, for example, in [11]. The simplest for-

mulation is probably to start by ignoring the inter-

conduction electron interactions and solving for the

eigenstates of the single-particle terms in the Hamil-

tonian (kinetic plus lattice potential energy); in this

way one generates the usual Bloch-wave states,which

are eigenstates not of total momentum P but rather

of total quasimomentum K, and are labeled corre-

spondingly by a quasimomentum index k and spin

 . The resulting system is the analog of the nonin-

teracting gas in the translation-invariant case. Then

one turns on the interaction and assumes that the re-

sulting groundstate and low-lying excited states can

be put into one–one correspondence with these of

the starting (“noninteracting”) state. It is important

to note that, because of the possibility of umklapp

processes,the interaction does not conserve the total

quasimomentum K; hence while one can still label

the resulting states by ın(k,  ), k no longer has the

physicalinterpretationof truequasimomentum (and

in fact the operator

K is nondiagonal in the ın(k, )

basis). This observation has some importance in the

context of the discussion of the “mid-infrared” peak

in the cuprates: see next section.

Is the normal state of the cuprates a Fermi liq-

uid (in the same sense as that in which the normal

state of

He can be said to be a Fermi liquid)? This is

one of the most hotly debated questions in the the-

ory of cuprate superconductivity,andto discuss it we

need to define it carefully. Let us first ask the ques-

tion: Suppose we could somehow cool the system

down to zero temperature while forbidding the su-

perconducting transition to occur. Would the result-

ing “groundstate” and low-lying excited states con-

form to the definition of a Fermi liquid? In the case

of liquid

He (where there is apparently no practical

way of suppressing the superﬂuid transition, so that

this really is only a thought-experiment) the answer

is almost certainly yes. In the case of the cuprates,

there are real-life possibilities of suppressing the su-

perconducting transition,in particular (a) by the ap-

plication of a sufficiently strong magnetic field and

(b) by doping the CuO

planes with certain types

of impurity such as Zn. Although for practical rea-

sons rather few systematic studies have been done of

the resultant state an answer seems to be emerging

which is rather intriguing [12]: There appears to be a

critical concentration p

∼ 0.19 of the hole-doping

parameter p (number of holes per CuO

units) such

that for p > p

the lowest-lying states indeed have

Fermi-liquid like character. However, to the left of

this point in the phase diagram (i.e. for p < p

)the

behavior appears to be qualitatively different, with

the system tending continuously towards an insulat-

ing state as the temperature falls towards zero.If this

is really the correct picture, then it is not entirely

clear that the question “does the normal state of the

cuprate superconductors behave as a Fermi liquid?”

has any clearly defined meaning. Of course one could

in any case argue that the system placed in a strong

(∼ 30–60 T)magnetic field, or doped substantially

with Zn impurities, is not “the same” system as a

true (superconducting) cuprate, and that therefore

no conclusions can be drawn from experiments of

the above type. In that case it is conceivable that the

true situation is the following: If one, by theoretical

fiat rather than by any realistically available experi-

mental means, could forbid the formation of Cooper

pairs in the (pure,zero-field) system while cooling it

to zero, then the “groundstate” and low-lying states

would indeed be of Fermi-liquid type, as in the cor-

responding thought-experiment with

He. However,

in both cases we should expect the Fermi-liquid de-

scription to break down badly when the temperature

becomes some (small) fraction of T

, let us say for

example 0.1T

.Now,inthecaseof

He, (T

∼

−2

) this“breakdown”temperature is still some way

above T

, so that there is a fairly wide temperature

regime (3mK

∼

50 mK) where a well-defined

Fermi-liquid behavior is seen. On the other hand, in

the cuprates the breakdown temperature could be

so little above the observed T

that no clear Fermi-

liquid regime is seen. It has to be said, however, that

this scenario, while perhaps plausible for the higher-

temperature compounds such as the T and Hg se-

ries,ismuchlesssoforcompoundssuchas(opti-

mally doped) Bi-2201 where T

is only in the range

10–30 K, and to defend it one has to argue that such

compounds are in some way untypical, even in the

normal state, of the cuprates as a whole.

A point which has been widely noticed is that

whileatoptimaldoping the resistivity ofthecuprates

24 Superﬂuid

He and the Cuprate Superconductors 1525

is rather accurately linear in T,it appears to cross over

to the T

form characteristic of Fermi-liquid theory

as one increases the doping p (“overdopes”). Other

experimental properties, where they have been mea-

sured, also appear to approximate better the Fermi-

liquidpredictionswith overdoping.This observation

appears entirely consistent with the hypothesis that

the crossover observed as a function of p in samples

where T

has been suppressed to zero is indeed char-

acteristic of the behavior of a (hypothetical) pure,

zero-field sample in the absence of pair formation.

Exactly what are the implications of this scenario for

the interpretation of the finite-T behavior as a func-

tion of p is at present a hotly debated issue.

24.3 Response Functions:

TheMIRPeakintheCuprates

One of the most universal properties of the super-

conducting cuprates (though it is also found in non-

superconducting ones) is the so-called midinfrared

peak in the optical spectrum.The most dramatic way

of bringing out this feature is to plot as a function

of frequency not, as is conventionally done, the raw

data (reﬂectance) or the (inferred) real part of the

conductivity, but rather the so-called loss function

L(!) ≡ −Im 

−1

(!)(where(!)istheq →0limitof

the complex transverse dielectric constant 

⊥

(q!)).

For a conventional metal described by the Drude the-

ory this quantity would have a narrow peak at the

plasma frequency !

, which is typically of the or-

der of a few eV, and for ! small compared to !

(but large compared to the scattering rate  )would

be very small. For a more realistic (band) model

L(!) should of course have features corresponding

to inter-band transitions,butunless the band gap E

is anomalously small there should still be a regime

 << ! << E when it is small. The experimen-

tal behavior of L(!) in the cuprates is very charac-

teristic and very different from these expectations.

L(!) is small below ∼0.1eV, but there is a wide and

strong peak extending from about that point to ∼

1–1.5 eV (the exact upper limit depends on the sys-

tem), at which point L drops precipitously almost to

zero, thereafter slowly increasing again (see, for ex-

ample [13]).

Because this “midinfrared” (MIR) peak occurs in

nonsuperconducting cuprates as well as the super-

conducting ones, it has often been regarded as irrel-

evant to the mechanism of superconductivity.How-

ever, already several years ago it was established [14]

that the transition to the superconducting state is ac-

companied by a change in the reﬂectance in the MIR

regime of the order of 1%, about two orders of mag-

nitude larger than expected in a naive BCS-type the-

ory, and recent ellipsometric measurements [15,16]

have strongly confirmed this result. Thus, while it is

at present unclear exactly what relation these phe-

nomena have to the mechanism of superconductiv-

ity, they can certainly not be dismissed as totally ir-

relevantto it.Thus it becomes a more urgentquestion

to understand the origin of the MIR peak.

It is tempting to regard this peak, along with the

linear-in-T resistivity, the anomalous T-dependence

of the Hall effect, etc., as evidence that the normal

state of the cuprates cannot be a Fermi liquid. I shall

now demonstrate,using the analogy with liquid

He,

that this argumentisnot convincing and that it is per-

fectly consistent to interpret the MIR peak as simply

the “incoherent background” on a Fermi-liquid pic-

ture. Since the quantity (!) is, apart from a factor

of !

−2

, just the (transverse) current–current corre-

lation function in the limit q → 0, it is necessary to

analyze the behavior of such functions in the Fermi-

liquid scenario.

Consider then some quantity A(rt) which may or

may not be locally conserved.Define the correspond-

ingresponse function

(rt,r



) in the standard way

(see [17]), take its Fourier transform 

(q!)with

respect to the variable r − r



and t − t



, and consider

the limit q → 0. It now turns out that in Fermi-liquid

theory onehasthe following result.If A(rt) is a locally

conserved quantity, then in the static limit the corre-

lation function

(q!) can be expressed entirely in

terms of the properties of the low-lying (quasiparti-

cle) states,and generally speaking has a rather simple

generic form (which however involves the values of

relevant Landau parameters F

, F

). In particular,the

region of ! for which there is finite spectral weight

(Im

(q!)=0) tends to zero as q → 0. Thus, in

the case of liquid

He, we expect that the correlation

functions of the three conserved quantities N (par-

1526 A.J.Leggett

ticle number), S and P will have no spectral weight

at finite frequencies in the limit q → 0. Experiments

on ultrasound properties appear consistent with this

expectation in the case of N. The correlation func-

tions of S and P are more difficult to measure. On

the other hand, consider some quantity such as the

spin current  which is not locally conserved (in the

case of spin current it is easily verified that it can be

changed in a collision,even though the total spin and

total current are each locally conserved). In this case

therewillbesomecontributionto

(q!)fromthe

quasiparticle states, and this contribution will have

zero weight at finite frequencies as q → 0. However,

there is also a contribution from nonquasiparticle

states, sometimes called the “background” contribu-

tion, even in the limit q → 0. Physically, the pro-

cess involved may be regarded as the “undressing”of

a quasiparticle by turning off part of its screening

cloud.Inthecaseofthespincurrentitturnsoutthat

one can actually calculate the relative contribution

of this background to the sum rule







(q!)

In fact we have

=1−

∗

1+F

. (24.6)

Since for

He at melting pressure the ratio m

∗

/mis

about 6, while F

is not very different from zero, the

contribution of the background is about 80% of the

whole!

Now let us turn to the cuprates. In this case, while

total particle number and spin are still locally con-

served, the current J is not, because of the effects of

the crystalline potential. One’s first instinct in this

situation is to re-express J as a sum of terms respec-

tively diagonal and nondiagonal in the band index,

and then to argue that the nondiagonal terms in-

volveinterbandtransitions.Thusthesewillhavezero

weight below the energy E corresponding to the

relevant band gap, while the diagonal terms can be

treated just like the (true) current in the translation-

invariant case and thus will have zero weight at finite

! in the limit of q → 0. However, this is not correct,

because in the presence of umklapp processes, al-

though the total quasimomentum K commutes with

the single-particle terms in the Hamiltonian, it does

not commute with the interaction. Hence the quan-

tity 

(q!) can have finite spectral weight in the

region !  E,eveninthelimitq → 0. (It should

perhaps be cautioned that “E” here should refer to

the band gap of the noninteracting system: in the

presence of strong interaction it is not obvious that

E even retains any clear meaning). Thus, there is

no a priori objection to fitting the MIR peak into a

Fermi-liquid picture.

24.4 The Cooper-Paired States of

Superﬂuid

He and the Cuprates:

General Considerations

As noted in Sect. 24.2, in the case of liquid

He there

is a regime of more than a decade in temperature

where the system appears to behave as a Fermi liquid

in the sense of Landau. Indeed, normal

He is often

regarded as the paradigm of such a system. This re-

gion is bounded below by the onset of superﬂuidity,

which is almost universally believed to be a conse-

quence of the onset of Cooper pairing. It is therefore

natural to try to construct a theory of the paired (su-

perﬂuid) state which as it were takes off from the

Fermi liquid description of the normal state. A sys-

tem so described is often referred to as a “superﬂuid

Fermi liquid”. Before doing so I very brieﬂy review

the most salient properties of the superﬂuid states.

For details see, e.g. [19], or Chap. 4 of [9].

Actually, what is most striking about the phase di-

agram of liquid

He below 3 mK is that the superﬂuid

“state” actually corresponds not to one but to three

different phases, which are conventionally known as

theA,B,andA

phases. The B phase is stable over

mostoftheP–Tplaneinmagneticfieldsoflessthan

a few kG; the A phase is stable, in zero field, at high

pressures and temperatures

∼

0.7T

, and also stable

in fields

∼

6kG for arbitrary pressure, while the A

phase occupies, in finite magnetic field, a thin sliver

of the phase diagram between the A (or B) and nor-

mal phases.The B phase appears to be isotropic as re-

gards most of its properties and its thermodynamics

24 Superﬂuid

He and the Cuprate Superconductors 1527

appear to be qualitatively similar to those of a classic

(BCS) superconductor.In particular, the data on the

specific heat are consistent with its vanishing expo-

nentially as T → 0. By contrast, the A phase shows

considerable anisotropy, e.g. in the ultrasonic atten-

uation, and its thermodynamics is different from the

standard BCS-like behavior. In particular, the spe-

cific heat appears to tend to zero as T→0asapower

law rather than exponentially. As regards the param-

agnetic susceptibility (which, remember, in the case

He is due entirely to the nuclear spins), this falls

rapidly in the B phase to a value as T→0ofabout

1/3 of the normal-state value, while in the A phase it

is independent of temperature and very nearly (but

not quite!)equal to the normal-state value. The prop-

erties of the A

phase are generally similar to those

of the A phase. It should be mentioned that although

these three new phases havebeen referred to fromthe

very earliest days as the “superﬂuid” phases of

He,

the evidence for superﬂuidity as such is very much

more circumstantial than in the case of

He. In fact,

had one not had theoretical ideas developed in the

context of

He and metallic superconductors to guide

one, it seems likely that the phenomenon would have

remained hidden for even longer than the 15 years

or so for which it went undetected in

He(He-II).

Let us now discuss how to formulate the the-

ory of Cooper pairing in a system such as

which above T

appears to be behaving as a normal

Fermi liquid. A simple semi-phenomenological way

to do this is as follows. We simply define operators

p

, ˛

p

which create and destroy Landau quasipar-

ticles rather than real atoms (e.g. when acting on

the interacting groundstate, the operator ˛

p







(|p |> p

, |p



|< p

) creates a state with ın(p )=+1

and ın(p







) = −1). These operators rather obvi-

ously satisfy Fermi statistics, and in general have the

same status in the context of the groundstate and

low-lying states of the interacting system as do the

original fermion operators a

p

, a

p

with respect to

the corresponding states of the noninteracting sys-

tem. Then, in addition to the “Fermi-liquid” terms

(24.3) in the Hamiltonian, which are diagonal in the

ın-representation, one introduces a BCS-like “pair-

ing” term of the usual general form. Formulated in

terms of the quasiparticle operators ˛

p

, ˛

p

one has

pair





˛ˇ ı

V (p, p



: ˛, ˇ,  , ı)˛

p˛

× ˛

−pˇ

−p





pı

. (24.7)

This term is actually a special case of the “scatter-

ing” terms which as already pointed out are left out

in (24.3). It is intuitively clear that the effect of such

a term is likely to be qualitatively similar to that of

the BCS pairing term in a Fermi gas. It will induce a

collective bound state of the Cooper type, but with

the individual fermion components of the pair be-

ing Landau quasiparticles rather than real fermions.

However, the internal configuration of the pair need

not have s-wave symmetry as in the classic super-

conductors. Rather, the effective angular spin and

angular momentum of the pair will correspond to

that“channel” in which the projection of the pairing

interaction V is most attractive (negative). A system

which is well described by this procedure is often

called a “superﬂuid Fermi liquid”(SFL).

It is clear that the role of the single-particle energy

inallthisisjustasintheBCScase,theonlydiffer-

ence being that the free mass m is replaced by the

effective mass m

∗

.What about the last term in (24.3),

involving theLandau interactionfunctionf (pp







which has no obvious analog in the BCS theory? The

crucial point is that, as discussed in [18], this term

only comes into play when there is some macroscopic

“polarization” of the system, and the formation of

Cooper pairs does not lead to such polarization (at

least in the approximation of “particle-hole symme-

try”.Thisisthoughttobeverygoodfor

He). Thus,

at first sight at least, the value of the transition tem-

perature T

should be a function only of m

∗

and the

coefficientsV in (24.7),and should be totally insensi-

tiveto theLandau f-function.Thisshouldalso be true

of these properties of the paired state (for example

the specific heat) which do not refer, either explicitly

or implicitly, to any macroscopic polarization; how-

ever,these properties (e.g.the spin susceptibility and

normal density) which do involve such polarization

should be sensitive to the Landau function,i.e. to the

relevant F

’s or F

’s. As an example, the ratio of the

spin susceptibility (T)inan(s-wave-paired)super-

ﬂuid state to its normal-state value 

which in the

BCS theory is just the Yoshida function Y(T), should

1528 A.J.Leggett

be given for a superﬂuid Fermi liquid by the expres-

sion

(T)





1+F



Y(T)

1+F

Y(T)

. (24.8)

Although this expression tends to 1 for T → T

and to zero for T → 0, its shape as a function of

temperature may be quite different from that of the

function Y(T). Similar remarks apply to the normal

density 

(T) as a function of T. For a much more

detailed discussion, see, e.g. [18], which uses the no-

tation Z

≡ 4F

At first sight,then the theory of a superﬂuid Fermi

liquidis nothing butthe theory of a superﬂuid Fermi

“gas” as described by BCS plus the set of molecular

fields associated with the Landau parameters F

, F

And indeed, it turns out that once the nature of the

pairing states is currently chosen and T

itself fitted

from experiment, such a description gives predic-

tions for the properties of the superﬂuid phase in

question as a function of T/T

which are quantita-

tively accurate typically at the level of a few percent

and often better. As we shall see, this observation is

actually quite puzzling, but let us first brieﬂy pause

to identify the relevant pairing states.

It is by now almost universally believed, on the ba-

sis of NMR and other experimental signatures, that

the pairing states corresponding to the observed A,

BandA

phases of

He all correspond to Cooper

pairs formed in a state with total spin 1 and to-

tal (apparent) orbital angular momentum 1 (i.e. a

p-state), but with different combinations of spins

and orbital angular momenta within this class. More

specifically, the Fourier-transformed order param-

eter (pair wave function) F

k,˛ˇ

≡a

k˛

−kˇ

 is in

the case of the A phase a product of a spin func-

tion which by an appropriate choice of axis can be

written 2

−1/2

(↑

↓

+ ↓

↑

), i.e. corresponds to

S =1, S

= 0, an orbital function which, by an

appropriate choice of the orbital axis can be writ-

ten as proportional to k

+ ik

,i.e.correspondsto

L = L

=1.TheA

phase has an identical orbital

wave function,but thespinfunction cannow be writ-

ten as simply ↑

↑

(S = S

= 1: only one spin species

undergoes pairing!). The B phase is a bit more sub-

tle; it is what in atomic physics would be called a

state, but with the spins rotated through a finite an-

gle relative to the orbital angular momenta.For more

details see Sect. 24.5.

The rather good agreement of the predictions of

the simple SFL picture with the experimentally ob-

served properties of superﬂuid

He is actually rather

surprising at first sight,for the following reason: Un-

like in the classic metallic superconductors where

there is a natural energy cutoff of the order of the

Debye energy for the pairing interaction

pair

,in

liquid

He not only is there no obvious such cut-

off, but we do not know a priori where the pairing

terms came from in the first place. Recall, this is a

pairing interaction between quasipa rticles,notbe-

tween real atoms, and we have very little quantita-

tive knowledge of the “structure” of a quasiparticle.

In any event, there seems no good reason why the

pairing interaction should not extend up to energies

where the BCS assumption of a constant density of

states breaks down badly, or why it should not it-

self depend substantially on the magnitude of the

momenta p and p



. Why do these complications not

affect the temperature-dependence of the measured

quantities?

The answer to this dilemma (or at least a large

part of it) was given in a classic 1961 paper [20]

by Anderson and Morel, which long predates the

experimental discovery of pairing in

He. Imagine

that the original inter-quasiparticle pairing interac-

tion indeed has the above properties, but that the

characteristic scale 

over which the strength of

V, the density of states, etc., varies is much larger

than the experimentally measured T

.Thenonecan

choose an intermediate energy cutoff 

such that

 

, and take into account the “scatter-

ing” into pair states of energy |  |> 

by a stan-

dard multiple-scattering (t-matrix) procedure. As a

result we get a problem where the pairing interaction

operates only within a “shell” of width 2

around

the (quasiparticle) Fermi energy,with a (single-spin)

density of states N(0)≡



d



and pairing interac-

tion V which is constant as a function of energy (i.e.

of | p | within this shell). This is just the original

BCS assumption generalized to include possible an-

gular dependence of V. Thus, all the unknowns are

buried in the quantities N(0) and V(ˆp, ˆp



). The tran-

24 Superﬂuid

He and the Cuprate Superconductors 1529

sition temperature T

is determined by N(0) and the

magnitude of the most attractive component V



V. The angular dependence of the order parameter

reﬂects the identity of this “most attractive”compo-

nent. Thus once T

and the symmetry is determined

everything else goes exactly according to the BCS

prescriptions and there are no further unknowns.

The above discussion leaves one major question

unanswered:Why should there be more than one sta-

ble superﬂuid phase, even in zero external magnetic

field? As mentioned above, all the evidence is that

both theA and B phases are spin triplet p-wave states

butwith different formsof the orderparameter.Such

a situation cannot occur in a simple BCS-type the-

ory with a single fixed value V

of the interaction in

the p-wave channel. In such a theory the B phase is

the unique free energy minimum. The generally ac-

ceptedexplanation,originallygivenbyAnderson and

Brinkman [21], is that because the effective interac-

tion between quasiparticles is strongly renormalized

and the renormalization process is itself sensitive to

the pairing state, the “effective” value of V

is slightly

different for the A and B phases, and this small dif-

ference is enough to stabilize the A phase, even at

zero field, in a small region of the P–T plane. This

effect is formally of order (T

)

and is a correc-

tion to the simple superﬂuid Fermi-liquid picture.A

comprehensive discussion of all corrections which

appear to this order has been given by Serene and

Rainer [22].

Now let us turn to the superconducting cuprates,

and suppose that we take the transition temperature

of a given compound from experiment and con-

sider the superconducting-state properties as func-

tions of the reduced temperature T/T

.Consider first

those bulk properties which within the BCS theory

are only indirectly sensitive to the energy gap (T).

Examples are the specific heat,spin susceptibility and

superﬂuid density which is measured in the standard

way by the penetration depth. Then the most salient

qualitative observation is that the general pattern of

the temperature-dependence of these quantities, at

least in the optimally doped and overdoped regimes,

is very similar to that expected in a superﬂuid Fermi

liquid.Indeed,if we restrict ourselves to“not too low”

temperatures (say T/T

∼

0.2),it is actually quite dif-

ficult to distinguish the system from a classic BCS

superconductor with s-wave pairing! The character-

istic effects associated with the“exotic” nature of the

order parameter mostly come into play only at the

lowest temperatures,see next section.For a reason to

be discussed in the next section,it does not make a lot

of sense to ask whether we can fit the temperature-

dependences at say the1% level by using a SFL model.

In the“underdoped”region of the phase diagram the

situation is less clear. For example, the specific heat

and the spin susceptibility show considerable “pre-

cursor” effects at temperatures above T

.Fromthis

alone it is clear that they cannot follow the SFL pre-

dictions.

If one considers phenomena such as angle-

resolved photoemission or tunnelling which, at least

in a BCS superconductor, are direct measures of the

energy gap (T), then even at optimal doping (or

in the overdoped regime) the agreement with the

SFL model is more problematic. The ratio (0)/T

inferred from this data is often much larger than the

BCS value 1.76, and in addition it is not always even

clear that the “gap” (T) inferred from these data

actually tends to zero in the limit T → T

.Thispat-

tern is amplified in the underdoped regime, where

a feature which looks suspiciously reminiscent of an

energy gap (the“pseudogap”) persists in the data up

to several times T

.Tosummarize,thedataseemto

be qualitatively consistent with an SFL picture at op-

timal doping and in the overdoped regime, but not

in the underdoped regime. This is not particularly

surprising, since we saw in Sects. 24.2 and 24.3 that

a similar statement could be made about the normal

phase.

It is interesting to ask,does there exist a quantita-

tive test of SFL theory in the cuprates? Recall that

inthecaseofnormal

He, the normal state does

permit such a quantitative test, see Sec. 24.2. Rather

serendipitously, it turns out that the answer is yes,

as pointed out by Chiao et al. [23]. They note that

according to recent theory [24], in a superconduc-

tor with the order parameter symmetry believed to

be characteristic of the cuprates and described by

the SFL model the low-temperature limit of the ther-

mal conductivity divided by T is independent of the

impurity concentration and a function only of two

1530 A.J.Leggett

parameters which are measured independently in

ARPES experiments. Thus, if the SFL model applies,

from the ARPES data it should be possible to predict

the thermal conductivity with no free parameters.

It turns out that such a prediction works essentially

perfectly foroptimally doped BSCCO, so that one can

say that the SFL model is at least not excluded for this

system. One cannot of course say that this proves it

to be right, since it is possible that the SFL model is a

special case of a more general class of models,which

makes the same prediction.In principle, further tests

of this quantitativetype are possible by using data on

e.g. the low-temperature specific heat,but the quality

of the data at present is inadequate for this. It should

further be noted that while the prediction of the SFL

model for the T → 0 temperature-dependence of

the penetration depth involves a Landau parameter

which cannot be independently measured, the value

required for this parameter in order to fit the data is

quite “reasonable”(i.e. ∼1, not ∼10 or 0.1!).

A final question of interest concerns the “size” of

the Cooper pairs, and the related issue of their pos-

sible “pre-formation”. Although the definition of the

“size” (or “radius”)  of the pairs is to some extent

a matter of convention, a natural choice is that it is

the mean square radius associated with the so-called

“anomalous average” F(r) ≡

(0)

(r) which

characterizes the paired state (the real-space Fourier

transform of the quantity entering (24.10) below, i.e.

the single macroscopically occupied eigenfunction

of ˆ

). For a simple BCS model  is then only weakly

temperature-dependent and of order of magnitude



/k

(i.e. equal, up to a numerical factor of

order unity, to the“Pippard coherence length” 

As is well known, for the classic superconductors

such as Al the quantity  is of order 10

–10

Å. This

is typically several orders of magnitude greater than

the interparticle spacing a. It is this state of affairs

which is the key to the remarkable success of the

BCS (mean-field) theory of superconductivity. An

equivalent way of looking at it is to note that the

ration a/

is up to a factor of order unity equal to

/

(∼ T

). The pair binding energy  is very

small compared to the Fermi energy 

, so that the

pair formation process affects only a fairly thin shell

of states around the Fermi surface. If on the other

hand it were to turn out that  is comparable to a

(or  or kT

to 

) then we shouldexpect ﬂuctuations

around the mean-field solution to play a major role.

In particular we might get a substantial range of tem-

perature where Cooper pairs are formed locally, but

do not undergo the “quasi-Bose–Einstein condensa-

tion” characteristic of the BCS state. The scenario is

sometimes referred to as that of “pre-formed pairs.”

In the extreme limit   a, the picture would be one

of tightly bound di-electronic (bosonic) “molecules”

which form at some high temperature T

and then

undergo Bose condensation at a much lower, unre-

lated, temperature T

BEC

 T

As remarked in Sect. 24.1, the ratio T

is con-

siderably larger for both

He and the cuprate super-

conductors than for classic superconductors such as

Al. If we work in terms

of the Cooper-pair radius

,thenitturnsoutthatfor

He  varies from about

200 Å at melting pressure to about 1200 Å at satu-

rated vapor pressure. This is large compared to the

interparticle spacing (∼ 3 Å). It is therefore not sur-

prising that superﬂuid

He appears to be very well

described by the mean-field BCS theory. In the case

of the cuprates,it is not entirely clear that we should

continueto usethe estimate  ∼ 

/mkT

.Ifhow-

ever we do so, and take the value of 

(for BSCCO)

from ARPES measurements, see, e.g. [23], we find

that at optimal doping  is typically ∼ 50 Å, while

if we take the relevant “interparticle spacing” to be

that of the extra holes doped into the CuO

planes it

is about 8 Å at optimal doping. Thus, the ratio /a is

only ∼6.Awayfrom optimaldopingit increases.If we

take this result seriously, it implies that while we are

by no means in the limit of “di-electronic”molecules,

we shouldexpect ﬂuctuationsto givesubstantialcor-

rections to the mean-field theory; such effects are in-

deed seen in the cuprates, most spectacularly in the

macroscopic electromagnetic properties. A particu-

lar prediction, as noted above, would be that there

should be a fairly substantial regime of temperature

where pairs are formed but are not Bose condensed.

In this regime the “local” properties of the system

Working in terms of a/ rather than T

has the advantage that the quantity a (≡ n

−1/3

where n is the relevant particle

density) is better defined than T

24 Superﬂuid

He and the Cuprate Superconductors 1531

should be similar to those of the superconducting

phase, but there should be no long-range order and

hence no superconductivity (in the sense of zero re-

sistance or a Meissner effect).A substantial school of

thought [25] claims that the so-called“strange metal”

regime of the cuprates which appears on the under-

dopedside of the phase diagram is exactly the instan-

tiation of this prediction, and in particular that the

approximately defined line of anomalies in the ex-

perimental properties usually denoted by T

∗

(p) cor-

responds to the onset of pair formation (which, just

as in the case of the formation of ordinary diatomic

molecules, is expected to be gradual rather than by

a phase transition). Whether such a scenario can ex-

plain the properties of the “strange metal” regime

quantitatively or even qualitatively is at present the

subject of vigorous controversy. Parenthetically, it is

amusing that followingearly suggestions along these

or similar lines for the cuprates, it has been pro-

posed [26] that normal

He, usually taken to be our

paradigmatic Fermi liquid, see Sect. 24.2, may itself

be described by a scenario of this general type.

24.5 Symmetry of the Order Parameter

Perhaps the most striking difference between super-

ﬂuid

He and the cuprates on the one hand, and the

classic superconductors on the other, is that while in

the latter case the order parameter is believed to have

the same symmetry as the lattice, in both

He and

the cuprates the general belief is that it is “exotic”,

that is, transforms according to a nontrivial repre-

sentation of the symmetry group G of the Hamilto-

nian. This may well also be true in the case of certain

heavy-fermion and organic superconductors.Forthe

moment, I will assume that in each case the order

parameter corresponds to a single irreducible repre-

sentation of G. This point will be brieﬂy re-examined

below.

In the case of liquid

He, where there is no lat-

tice, the symmetry group is (to the extent that we

ignore the small nuclear dipole-dipole interaction) a

product of the group SO(3)

orb

of rotations in orbital

coordinate space and the group SU(2)

correspond-

ing to rotations in spin space; we need SU(2) rather

then SO(3) because of the double-valued character

of the spin functions. For a complex (Cooper pair)

of two particles of spin 1/2 the relevant irreducible

representations of SU(2) correspond to a singlet and

a triplet,and on the basis of the behavior of the spin

susceptibility it is almost universally believed that

both the A and the B phases correspond to the triplet.

Under all but very special circumstances, the Pauli

principle then implies that the orbital wave function

is odd-parity, i.e. corresponds to “angular momen-

tum”  =1, 3, 5 ... All the evidence is that in both

A and B phases  is equal to 1. Thus, in both the A

and B phases of

He the Cooper pairs have  = S =1.

Of course, this statement does not characterize the

pairing state uniquely, since we have three possible

Zeeman substates corresponding to spin and three to

orbital angular momentum,thus nine possible states

in all. The most compact characterization of the or-

der parameter turns out to be through the quantity

d(k) ≡ (i

 )

˛ˇ

a

kˇ

−k˛

, (24.9)

where the components of  are the Pauli matrices.

The quantity d(k) automatically behaves as a vector

in spin space, and because of the  = 1 requirement

must also transform as a vector in orbital space. The

formsof d(k) inferred from NMR and other evidence

to correspond to the A and B phases of

Heareasfol-

lows: For theA phase d(k) isgivenupto a multiplying

constant by

d(k)=

d(k · (ˆn

+ i ˆn

)) , (24.10)

where

d is a real vector in spin space and ˆn

and ˆn

are

a pair of mutually orthogonal (real) unit vectors in

orbital space. It is conventional to define a third unit

vector

 by

 ≡ˆn

×ˆn

.The physical interpretationis

that in theA phase the pairs have total spin 1 and spin

projection 0 on the (spin-space) axis

d,andinorbital

space have “apparent” angular momentum 1 along

the(orbital-space) axis . To discuss the B phase, it is

convenient to start from the state d(k)=k.Inthis

state the apparent orbital angular momentum is al-

ways directed oppositely to the spin,so this is what in

the language of atomic physics would be denoted as

state.The actual state of

He-B is believed to be

obtained by rotating the spin coordinate system rel-

ative to the orbital one through an angle  around an

1532 A.J.Leggett

axis ˆ!. To minimize the dipole–dipole energy  has

to be cos

−1

(-1/4) ≈ 104

◦

, but the axis ˆ! is fixed only

by higher-order effects. The resulting state, while it

has L = S = 0, has a finite value of the quan-

tity L × Sdirected along ˆ!. For further details see,

e.g., [9].

Inthecaseofthesingle-planecuprates,itismost

natural to regard the pair wave function as de-

fined within a single CuO

plane, with transfer be-

tween neighboring planes effected by a Josephson-

type mechanism. Then to the extent that we neglect

spin–orbit interaction, the relevant symmetry group

is the product of the orbital symmetry group, which

is exactly or approximately that of the square, C

times the group SU(2)

spin

. Again, the possible spin

representations are singlet or triplet, and the evi-

dence from the NMR behavior strongly favors the

singlet. This then suggests that the orbital state has

even parity. The even-parity irreducible representa-

tions of C

are four in number, and are denoted

in the formal notation of group theory A

and B

, or informally s (or s

), g(or s

−

), d

−y

and

, respectively. There is by now rather convinc-

ing evidence (see, e.g. [27]) that, at least in these

cuprates where phase interference experiments have

been done (YBCO, T-2201, Bi-2212...) the symme-

try of the order parameter is d

−y

). There is a

strong suspicion that this is so for all the supercon-

ducting cuprates.

Here it is necessary to emphasize an important

distinction: The orbital symmetry group SO(3) of

He(aswellasthespingroupSU(2)

) is continuous.

Thus, once one has specified not just the irreducible

representation involved (in this case  = S =1)but

also the“nature”of the state (e.g.A-like or B-like),the

only freedom left over is the “orientation” in orbital

and/or spin space, e.g. for an A-like state the vectors

 and

d. It is clear that the free energy must be inde-

pendent of those in the neglect of the small nuclear

dipole–dipole interaction. This has the consequence

that if onemakes the hypothesis that the SFL model is

exact for

He and chooses a particular state,e.g.A or

B, one can calculate the free energy and most of the

relevant experimental quantities exactly as a func-

tion of T/T

and of the Landau parameters. It thus

makes sense to try to quantify the small discrepan-

cies from these predictions,cf., e.g. [22]. By contrast,

the orbital group C

of the cuprates consists of dis-

crete operations (/2 rotations and reﬂections), so

the statement that the symmetry of the order pa-

rameter is d

−y

tells us only that it changes sign

under /2 rotations and also under reﬂections in a

◦

crystal axis. This still allows a wide variety of

dependencies on k,fromasimpleformtoacompli-

cated form with several nodes other than the 4 in the

◦

direction which are imposed by symmetry alone.

Consequently, the statement that the system behaves

as a SFL does not uniquely determine the free en-

ergy and other quantities as functions of T/T

and

the Landau parameters: to obtain such a result one

needs a specific microscopic model whichfixesquan-

tities such as the single-quasiparticle energy and the

effective pairing interactions as detailed functionsof

k as the latter varies around the Fermi surface.

A second difference associated in part with the

different structure of the groups SO(3) and C

that in the former case “continuous” mixing-in of

irreducible representations other than the principal

one allowed, whereas in the latter this requires a

second phase transition (on this point, see Sect. 3

of [28]). Thus the absence of any observed second

phase transition in the essentially pure tetragonal

crystal T-2201 is strong evidence that the bulk or-

derparameter is pure d

−y

,whilethepairingstateof

He-A (though not

He-B) may have a small admix-

ture of other odd-parity orbital states ( =3, 5 ...).

In practice this seems very unlikely to produce quan-

titatively significant corrections.

In general, the fact that the order parameter of a

given system is“exotic”(i.e.has symmetry lower than

that of the Hamiltonian) has a number of interesting

consequences. First, in most such cases the order pa-

rameter, and hence the energy gap which is crudely

speaking proportional to it, will have nodes at one or

more points on the Fermi surface. This means that

the number of normal quasiparticles excited at low

temperatures will be proportional to a power of T

rather then exponentially small. Consequently, ex-

For cuprates such asYBCO with appreciable orthorhombic anisotropy the analysis becomes rather messier,see e.g.,[28].