HIYSICA
Physiea C 185189 (1991) 120129 NorthHolland
Spin Fluctuations and High Temperature Superconductivity in the Antiferromagnetically Correlated Oxides: YBa2Cu307; YBa2Cu306.63; Lal.85Sr0.15CuO4 David Pines Physics Department, University of Illinois/UrbanaChampaign, 1110 W. Green Street, Urbana, Illinois 61801 The evidence from NMR experiments for the existence of strong temperature dependent antiferromagnetie correlations among nearly localized Cu2+ spins in the cuprate oxide planes is reviewed. It leads naturally to a description of the normal state as a nearly antiferromagnetie Fermi liquid in which the magnetic excitations are commensurate, or very nearly so, antiferromagnetic paramagnons whose spectr~,m at low frequencies can be obtained from fits to NMR experiments, and whose coupling to quasiparticles on a 2D square lattice is responsible for the measured anomalous transport and optical properties. Recent work by Monthoux, Balatsky and Pines demonstrates that the retarded quasiparticle interaction induced by these paramagnons can lead to a superconducting state with dx2_ y2 symmetry, a high transition temperature, and energy gap behavior comparable to those measured. NMR experiments on the superconducting state are shown to support dwave pairing, while existing penetration depth experiments do not appear to provide conclusive evidence concerning the pairing state. 1. INTRODUCTION In this talk I describe a ~bottomup~
first principles. ''(2), there are a large number of
experimentally based approach to understanding
novel properties from which to choose. Of these,
the cuprate oxide superconductors, in which a
the low frequency magnetic properties measured
model Hamiltonian for microscopic calculations
in NMR experiments are perhaps the most
is constructed and constrained from a
unusual, so I begin with these. Experiment(3)(4)
phenomenological description of the normal state.
shows that a single, nearly localized, Cu 2+ spin
This was the approach which John Bardeen(1)
component is responsible for both the measured
used in the course of developing the microscopic
Knight shifts and spinlattice relaxation times. A
theory of superconductivity with Leon Cooper and
phenomenological theory makes evident the
Bob Schrieffer, and in opting for this approach, I
presence of strong temperaturedependent
have been s, ~ ea~,y ^*'  im~uenced " ~u~ nlylong and close
¢:LILt, LL~.,AZVLJLI¢.Z~AA~.,~,.LI..,
association with John.
spins which lead to a substantial peaking of the
Almost every aspect of the normal state
~V~LJL~Af,.~I,A't.~A~,.~
a~*..,L, TVV,.,'.~.AI
~AA't.,
~..~'r..A
low frequency spinspin correlation function, ~"
behavior of the cuprate oxides is anomalous, so
(q,¢o), for wavevectors near (~Ja, rJa) and a very low
that if one is to follow Richard Feynman's dictum,
characteristic energy, scale.(5) These results led
"It does not make any difference what we explain,
me to suggest that the normal state is a nearly
as long as we explain some property correctly from
antiferromagnetic FelTni liquid of coupled
Elsevier Science Publishers B.V.
D. Pines / Spin fluctuations and high temperature superconductivity quasiparticles and antiferromagnetic paramagnons, and that the spin fluctuation induced quasiparticle interaction might be the physical origin of the transition to the superconducting state.(6) In Sec. 2 I discuss NMR experiments and in
121
temperature dependence, one finds instead 17W(T) ~ C2a Xo (T) T F (1) where F is a magnetic Fermi energy (which may be weakly T dependent) and Ca, the transferred hyperfine coupling between the oxygen nucleus
Sec. 3 give a brief progress report on the properties
and a Cu 2+ spin localized at the nearest neighbor
of the two dimensional nearly antiferromagnetic
Cu sites, can be determined from Knight shift
Fermi liquid. Sec. 4 is devoted to a report on recent work with Philippe Monthoux and Alexander Balatsky (MBP)(7), which demonstrates that the retarded quasiparticle interaction induced by paramagnons leads uniquely to a superconducting state with dx2_y2 symmetry. With a quasiparticle paramagnon
m e a s u r e m e n t s . (9) This unusual behavior, was anticipated by Millis, Monien, and Pines.(5) •
The 63Cua spinlattice relaxation rate, on the
other hand, is greatly enhanced over what might be expected from the hyperfine coupling constant determined by the measured Knight shifl~, and displays markedly nonKorringa behavior. It may
state transport and optical properties, high
be expressed in the form 63Wa (T) (A~ff)2Xo (T) T ~ ~'SF (T)
transition temperatures and energy gap behavior
where ACeff,a material constant independent of
comparable to those measured are obtained (7). In
temperature and hole content, takes into account
Section 5 I discuss the extent to which present
both the anisotropic direct hyperfine coupling, Aa,
NMR and penetration depth experiment support
of an onsite Cu 2+ spin, and the nearest neighbor
the calculated dwave pairing.
transferred hyperfine coupling described by the
2. ANALYSIS OF NMR EXPERIMENTS ON THE
MilaRice Hamiltonian.(lo) In this notation, the
coupling consistent with the measured normal
NORMAL STATE The key results of NMR measurements on the normal state3,4, 8 may be summarized in the
(2)
low energy spin fluctuation energy, ~SF (T), is F ~SF (T) = RAF (T) (3) where RAF (T) is the temperature dependent
following way: antiferromagnetic enhancement of the 63Cu • For YBa2Cu307, where the uniform spin relaxation rate. From fits to experiment, one susceptibility Xo is temperature independent, th~ finds at 100I~ (RAF = 8, e~SF = 14meV) for 170 relaxation rate, 17W(T), displays Korringalike YBa2Cu307; (RAF = 23, ~SF = 5.6meV) for temperature behavior (17W(T) ~ T), but for other 1YBa2Cu306.63; and RAF = 40, ~SF = 3.2meV) for 23 oxygen concentrations, and for the 214 system, where Xo displays considerable
La 1.85Sr0.15Cu04.
D. P/nesI Sp/n J~ctuat/o~ and h/gh temperature supe~com/ucari0,
122
* The temperature dependence of ~0SF(T) may be
second represents the contribution from the
obtained directly from 63Cu relaxation rate and
antiferromagnetic paramagnons of energy
Knight shift measurements, since from Eq. (2), 63T1 (T) 63KS(T) T  ~'SF (T).
(4)
Quite generally,although neither 63T1 CI0 or 63Ks (T) [the spin part of the Knight shift,for which one could equally well substitute 17Ks (T)] display a simple variation with temperature, their product with temperature does; above 125 K it takes the CurieWeiss form, ~OSF(T) = a + bT.
~SF(T) = ~
~2 ;
(8)
is the a f correlation length which measures the range of the ay buildup of the static susceptibility at Q=(rJa, ~/a), ZQ (T) = Xo (T) (~2/~2o). In the MMP fits to experiments on YBa2Cu307, (5) YBa2Cu306.63,(11) and Lal.85Sro.15Cu04, (12) ~o is taken to be temperature independent and . 0.56a, while
(5)
The presence of a m a x i m u m in a plot of [63W(T)fP]
~2(T) const a s = c+dT
(9)
for, say, YBa2Cu306.63 thus reflects a temperature
is responsible for the measured temperature
region in which Xo (T) varies more rapidly with
dependence of the antiferromagnetic correlations
temperature than does the comparatively
(and the low characteristic spinfluctuation
smoothly varying ~SF (T).
energies, O)SF [T] or COSF [T]). If,on the other
A quantitative fit to the NMR experiments m a y
hand, one takes (~Ja) I and independent of
be obtained by combining the MilaRice
temperature, as RossatMignot has proposed from
Hamiltonian with the MMP ansatz for the
his fitsto neutron scattering measurements, 13
imaginary part of the spinspin correlation
then ~o must be both temperature dependent and
function. One obtains a W(T) lira Y ah2(q) X" (q,co) T = co~o q co
quite small compared to the latticespacing, a, in (6)
order to fit the N M R experiments on 63W(T). One can fit,63Wc(T), equally well with either choice of
where c~ refers to the nuclear site in question, parameters and, as shown by Millis and ah2(q) is the corresponding form factor (which includes the appropriate hypeffine coupling
Monien,(14) it is likelythat for YBa2Cu307 one can also arrive at reasonable fitsfor 17W(T), 89W(T),
constants), and X" (q,co)= x Xo (T) co )Co(T) (~2~o2) (c4COSF) [1+ (Q_q)2 ~212+ co2/co2SF F (7) The first term on the rhs of Eq. (7) is a "local moment" version of the usual Fermi liquid term; F, the magnetic Fermi energy, is  0.4 eV and is very weakly dependent on temperature. The
and the 63Cu N M R relaxation rate for a field applied in the ab plane, 63Wa(T). Matters appear otherwise for attempts to fitthe latterquantities for YBa2Cu306.63, or La1.85Sro.15CuO4, where the measured very much larger antiferromagnetic enhancement, RAF (T), makes it difficultto avoid "leakage" of the nonKorringa behavior of 63Wc(T)
D. Pines / Spin fluctuations and high temperature ~q,ercondu~vitv
12,~
on to 17W(T), which, in fact, maintains the "quasi
explore the influence of these excitationson the
Korringa" local m o m e n t behavior represented by
transport and opticalproperties of the normal
Eq. (1) for both materials.(9).(ls)
state we m a y followAnderson(Is)and use a one
A similar problem exists with the spinspin
band description of the planar excitations of a
correlation function obtained from the analysis by
2D square lattice which are assumed to satisfy
Cheong eta/. (16) of their neutron scattering
Luttinger's theorem; however, instead of
experiments on Lal.85Sro.15Cu04; Thelen and I(17)
introducing spinons and holons we assume that
find t h a t the degree of incommensurability which
in the absence of coupling to the spin excitations
they propose is so large t h a t the resulting
the quasiparticles obey the usual tightbinding
"leakage" of their peaks to the oxygen sites would
dispersion relation, ek = 2t (coskxacoskya) and
produce a behavior substantially different from t h a t measured for 17W(T) by Reven et a/.(15) The
describe their coupling to the spin excitations by Z ~ (~) ~ (~). ~ (_~) (10) Hint = ~1 ~,
message from the NMR community to the neutron
1 where ~ (q) = ~ Ya.l;.k tP+k+q,
scattering community is clear; the acid test of a
is the spin fluctuation operator whose correlation
suggested svinsvin correlation function is getting
function X (Qq,co) is taken to be strongly peaked
the right Quantitativebehavior ofl7W(T) not getting
around i~ = (g/a, g/a) with an energy spectrum
the right temperature dependence for 63Wc(T)!
which yields quantitative argument with NMR (or
Finally, it should be noted that one is only
neutron) experiments, and ~ (q,t) is a coupling
obtaining information about the very low frequency part of X" (q,o~)in N M R
O~c~5tlJk~ and "~ (q)
constant which may in general be both q and T
experiments; as dependent. Because ~ is commensurate while the
discussed by Millis and Monien, (14)one could, for example, replace ¢~2/c02SFin Eq. (7) by a quantity which has the same low frequency behavior, but a
Fermi surface is incommensurate, the resulting quasiparticle self energy, Y~(P,co, T) is sensitive to
larger cutoff. In this case the mean
the position of the quasiparticle on or near the
paramagnon energy would be increased.
Fermi surface (quasiparticles capable of coupling
3. IS T H E N O R M A L FERROMAGNETIC Both N M R
STATE A NEARLY
ANTI
F E R M I LIQUID?
and neutron scattering experiments
tell us that the dominant magnetic excitations for the planar Cu 2+ spins lie near Q = (rJa,g/a), and
to ~ will clearly have their properties modified co_n_s~_deroblymore than those which do not). and to co and T through the dependellce of X (Qq,co,t) on the latter quantities. Thus far a complete self consistent calculation
that their characteristic energy is small compared
of the resulting quasiparticle spectrum has not
to the magnetic Fermi energy, r ~ 0.4 eV. To
been carried out. However in weak coupling
D. Pines ] Spin fluctua'.kms and high temperature superconductivity
124
calculations using the MMP expression for x(Q
mechanism has, however, been rejected by many
q,o))(19), or an RPA expression (20) or an SCR
on the grounds that the likely transition
expression of the kind Moriya has described at this meeting(21), and taking the Fermi surface to be nearly circular, all authors find that the Fermisurface averaged imaginary part of the self energy
pairing state would be a "dwave" state, (see, however, Schrieffer et a/.(23)), in apparent contradiction with measurements of the temperature dependence of the penetration
takes the form < I m Y.(p,co,T)> ~ M a x (co,T)
temperature would be low, and that the likely
(II)
in agreement with resistivity and optical experiments and in striking contrast to the Fermi liquid value, Im Y.  (co2+ ~ T2). Philippe Monthoux in Urb:ma is in the process of implementing a selfconsistent calculation of Y. (p,e) which reflects the expected asymmetry in matching the Fermi surface to ~ in certain directions.(22) O n averaging over the Fermi surface, he obtains results of the form, Eq. (11). It remains to be seen whether the temperature dependence of Xo reflects selfenergy effects or the presence of a spin pseudo gap. 4. S U P E R C O N D U C T I V I T Y In view of the similarities between heavy electron superconductors and the cuprate oxides (both are strongly correlated electron systems which exhibit antiferromagnetic behavior), and
depth.(24) Since, however, the NMR experiments establish the existence of antiferromagnetic paramagnons, and these provide, as we have seen above, a plausible candidate for the anomalous normal state behavior, it seemed worthwhile to look again at the possibility that these might give rise to high temperature superconductivity. Monthoux, Balatsky, and I have begun such a reexamination(7), and I give here a brief report on the results of the calculations. MBP consider the first order spin fluctuation exchange between quasiparticles described by the interaction, Fig. (1); they use a spin fluctuation spectrum determined by fits to NMR experiments, and find, in common with the authors ci~d
above, (23) that this leads uniquely to a superconducting state with dx2_y2 symmetry, with an energy gap, A(k) = Ao(k) [coskxa  coskya] .
(12)
the m a n y indications that the heavy electron systems are unconventional superconductors for which spin fluctuations provide the superconducting mechanism, the spin fluctuation induced interaction quasiparticle has for some time been considered an appealing candidate for superconductivity in the cuprate oxides.(23) The
From plow of its dependence on the respective energies of the pairing quasiparticles, as shown in Fig. 1, they find that the characteristic energy scale which the spin fluctuation induced interaction is effective is  Fin 2, where F, the magnetic Fermi energy, is  0.4 eV; for realistic values of g2eff (Tc) they find transition
D. Pines / Spin fluctuations and high temperature superconductivity
{2 5
t e m p e r a t u r e s comparable to those measured.
41
...................
!
"YBoz C~ Oe.¢,,~
s,.o,o 1
..............
,j 3
.
A
40.
~~"2
"~"~,
, ~  4t
""..

"'.,,,,,, .....
~ .....".. ' "
~..
""' '"'.
".....
~,,~
..,,. ID 2.O 5JO I/~;f (Tc)Xo (TcJ (eV"t) •
o:2
'
o'.,~
'
o~s
'
~
'
T/To • "1"
0,0
• 0.1
0.1
",
....
 ....
0.2
, ....
i ....
0.3
, ....
i ....
. ....
0.4
0.$
p
FIGURE 1 S t r u c t u r e of the retarded interaction between quasiparticles calculated by Monthoux, Balatsky, and Pines. (7) For a quasiparticle on the Fermi surface (E' = 0.25 ev), the maximum interaction occurs for quasiparticles of energy E = 0.25 ev, which are coupled to E' via Q.
FIGURE 2 The t e m p e r a t u r e dependence of the maximum value of the energy gap for YBa2Cu307 (Te = 95K), YBa2Cu306.63 (To = 60K), and Lal.85Sr0.15Cu04 (To = 40K); the inset gives a plot of the relationship, F(Tc) 1 T¢ = ~ exp ( ): ~geff (To) Xo(Te)N(O) (a = 1.66, T1= 1.07) for YBa2Cu307; (a = 1.49, TI = 1.07) for YBa2Cu306.63; (a = 1.51, TI = 1.21) for Lal.85Sro.15CuO4 (from Monthoux, Balatsky and Pines(7)).
As shown in Fig. 2, MBP were able to fit their computergenerated transition temperature to an In the superconducting state, because the energyexpression of the BCS form r 1 Tc a ~~exp  k(Tc) =
dependent effective interaction depends on the (13)
energy gap, there is a feedback effect which MBP
where the dimensionless effective coupling
find leads to novel properties for A(T); as may be
constant, ~.(Tc) = TI g2eff (Tc) Xo(Tc) N(0), varies
seen in Fig. 2, even for weak to intermediate
from 0.48 to 0.33 depending on the compound, and
coupling strengths, the enera~v _~apovens uv very
a and T1 are material constants of order unity.
ratfidlv below T~, and reaches a maximum value which is large comvared to kT¢, in good . . . . ~¢;xxx~aal, .. * with cxper]ment. The q t l a l.... , ~ l v ": . .¢:. .
126
D. Pines
/ Spin fluctuations and high temperaturesuperconductivity
anomalous normal state transport and optical
properties. Moreover, since superconductivity in
properties.
high Tc compounds is three dimensional, interplane coupling will establish the true 3D i
i.z.
I
 ,
I, I,
It II ." iIIS
1.0"
I
i
. , . I . . 1: II II ::ill ii ." It ~:l~ : i
.~ll ,ll '. l l l.l , ~ , q '
~O.e
'.... Jj, ll
:=
i I "!
~v
asL
.4
." ,7
I.
~o,4L "~ I
/ /
.* .'.J 0~']~, (, , . , . o a z ~ 4 ( 1 6 o~l.,
~ =.~. ~ .  . ." I t
~i . ~ ~ ' " . ~ "
T/To
i x 'i,
=o,e 3
'
:
......... .... T,T,:O.,O
,':

Z 0.4
:
I I
/#.:,f
: # " "" : /' '//I /#' #' i ~ /' i ' " Z #\:j~?" ~.,r , 0,0(~ ' ,
0.2
"
io
I
I
.20 2 c,i/B
__,,,, o.= ........ T/T© = 0.50 T/T, = 0.70 . . . . . . TIT¢ • 0.80 IT/T©'0.90 T/T¢ • 0 . 9 4 ........ T/T¢ • 0.98 T/T,, • 0,99 I I i
3o
4o x iO"3
coherence of the order parameter. However, since the interplanar coupling is weak, it is to be expected that the inplane symmetry of the gap will remain dx2_y 2, so I turn now to the experimental evidence concerning the pairing state.
5. IS T H E P A I R I N G S T A T E D  W A V E ? In trying to understand the nature of the superconducting state, it is again useful to begin
FIGURE
3
The density of states for the dwave gap A(~) as a function of the variable (2o~B), where B = 2eV is the bandwidth at various values of (TPrc) for YBa2Cu3OT; the inset shows the calculated Knight shift in the superconducting state. The dotted line is the base value, the full line with the Fermi liquid correction, Foa = 0.6. The dots are the
experimental points of Barrett eta/.(28) (from Monthoux, Balatsky and Pines(7)).
with N M R
experiments which possess the great
advantage of probing the local surroundings of a given nucleus, so that these can be used to isolate the properties of the "planar" portion of the Fermi surface. Experiment(3)(4)(25) shows: * No HebelSlichter peak in either 63W(T) or 17W(T) in the vicinity of Tc
As M B P point out, although the qualitative
• A very rapid falloffof Xo (T) below Tc
agreement with experiment is encouraging, these
• A very rapid falloffof63Wc (T) below Tc
calculations represent only a firststep toward the
• A n approximate, but not identical, falloffof
development of a consistent theory of the
17We(T) and 63We(T) below Tc
properties of the normal and superconducting
• [63Wc(T)Pr] ~ Xo(T), not IN(O)]2. (26)
state. For example, before a quantitative
These results are consistent with dwave pairing
comparison with experiment on the transition
and antiferromagnetic correlations which do not
temperature and gap properties can be made, it is
very appreciably with temperature below Tc.(27)
important to incorporate lifetime effects and to
For example, Monien and I found it possible to fit
carry out a selfconsistent calculation of the
the temperature dependence of both 63W(T) and
chemical potential and its variation with doping,
Zo(T) with a large energy gap and dwave
which is needed as well as for the normal state
pairing(27), while M B P find a good fit to the N M R
D. Pines / Spin fluctuations and high temperature $uperconductivio.
127
experiments of Barrett e t a / . (28) can be obtained by
low temperature increase cannot occur, and
applying a Fermi liquid correction to their
would only be explicable as a new low temperature
calculated value of Zo for YBa2Cu307 (Fig. 3).
orbital relaxation process which somehow mimics
Added support for dwave pairing comes from experiments involving the substitution of nonmagnetic atoms in planar Cu sites. Thus Xiao et a/. (29) find that substituting 2% Zn in Lal.85Sro.15CuO4 reduces Tc from 40K to OK, while as discussed here by Kitaoka, (4) a comparable substitution of Zn in Yba2Cu307 leads to gapless behavior.
the effect found naturally in the dwave BulutScalapino calculation.
'i
,
i
i
•
.
i
1¢)
2
While all of the above results are easily explained by =dwave" pairing, an ardent advocate of =s'wave pairing could construct an explanation
I 02
*
I 0.4
°
I O.O
~
T 0.6
*
r/t.
for most of them. What appears, however, to be a "smokinggun" for dwave pairing comes from the experimental results on the anisotropy of 63W(T); found originally by Barrett eta/. (30), confirmed by Takigawa et al. (31), and discussed by Slichter(25) at this meeting, [63Wa(T)/63Wc(T)] displays an initial rapid drop below Tc of some 20% (from its
FIGURE 4 The anisotropic relaxation rate ratio for YBa2Cu307 as a function of (T/Te). The dots are the experimental values o~ Barrett eta/.(30), Takigawa et a/.(31) and Martindale et a/.(30); the line is the dwave pairing calculation of Bulut and Scalapino (32) (from Bulut and Scalapino(32)). These results led Thelen, Lu, and me to
temperature independent value of3.8 above Tc),
reexamine the information on the pairing state
followed by a gradual rise, until at low
provided by experiments on the temperature
temperatures it is some 30% above its normal state
dependence of the penetration depth, from which
value (Fig. 4). As the calculations of Bulut and
it is straightforward to extract pn(T)/p.(26) Annett,
Scalapino (32), which are compared with
Goldenfeld, and Renn (33) had earlier shown that a
experiment in Fig. 4, show, this result is uniquely
careful analysis of the experiments of Fiory et
explicable by the interplay of antiferromagnetic
ai.(24) on YBa2Cu307 leads to a T2 dependence for
correlations and dwave pairing.
[pn(T)/p] at low temperature (inconsistent with s
The rapid drop near Tc is produced by dwave coherence effects (remember that the gap opens up rapidly), while the low temperature increase results from the scattering of excitations between nodes of the gap function. With sstate pairs, this
wave pairing, possibly consistent with dwave pairing when Fermi liquid effects and impurity dwave scattering are taken into account). Thelen et al. (26) compare the results of FioD" et al. ~ith the results of }~sr experiments and with a dwave
128
D. Pines / Spin fluctuations and high temperature ~e.rconductivity
pairing calculation in which Amax (0) = 3.5 RTc and various Fermi liquid corrections are applied. As m a y be seen in Fig. 5, the different experiments do not agree, and the dwave pairing calculation is "representative"of the range of the different experimental results which are not at present consistent with one another. From this I conclude that penetration depth of experiments do not at present provide convincing evidence for or against either s or dstate pairing. 1
0.8
o;/J.,"I
0,6
0.4
0,2
0
0.2
0.4
0.6
0.1
I
FIGURE 5 Experimental measurements of [pn(T)/p]by Fiery et a/.,(24) Pumpin et al.(24) and Harshman eta/.(24) are compared with dwave pairing calculations for Amax (0) = 3.5 kTc using various Fermi liquid corrections (from Thelen, Lu, and Pines(26)). 6. C O N C L U S I O N As M B P note, Schrieffer and his colleagues(23) share our view that af correlationsplay a key role in determining the superconducting transition, but a.,~.!ethat the coupling between quasipa~ic!es and spin fluctuationsis so strong that the normal state excitations are spin bags; they focus on the secsnd order spin fluctuation exchange to obtain swave superconductivity, rather than using the firstorderterm to get dwave pairing. In the M B P approach, the results of experiments on the normal state may be used to
fix to fundamental quantities which enter the gap equation. W e are encouraged that our numerical results agree qualitativelywith m a n y experiments, and have very recently been confirmed by Millis(34) in an approximate analytic calculation which also includes selfenergy effects. ACKNOWLEDGEMENTS I take this occasion to remember John Bardeen, not only for many stimulating conversations on these and related topics,but for being a constant source of inspiration and encouragement during the thirtyone years we had officesnext to one another. I should also like to thank m y collaborators and colleagues, E. Abrahams, A. Balatsky, IC Bedell, J.P. Lu, A. MBlis, H. Monien, P. Monthoux, J. R. Schrieffer, C. P. Slichter,and D. Thelen for stimulating discussions on these and related topics, the National Science Foundation for its support, through grants DMR 8817613, DMR 8920538, and DMR 8809854 through the Science and Technology Center for Superconductivity, Andy Millis for a critical reading of this manuscript, and the Aspen Center for Physics for its hospitality during its preparation. REFERENCES 1. J. Bardeen, Encyclopedia of Physics, Vol. 15, p. 274 (SpringerVerlag, 1956). 2. R.P. Feynman, Rev. Mod. Phys. 29 (1957) 205. 3. H. Yasuoka, these proceedings. 4. Y. Kitaoka, these proceedings. 5. A. Millis, H. Monien, and D. Pines, Phys. Rev. B ~42,(1990) 167. For closely related RPAbased calculations see N. Bulut et aL (20), and SCR calculations, see T. Moriya, Y. Takahashi, and K. Ueda, J. Phys. Soc. Jpn. 52 (1990) 2905. 6. D. Pines in High T~mgerature Superconductivity, eds. K. S. Bedell, D. Coffey,
D. Meltzer, D. Pines, and J. R. Schrieffer (AddisonWesley, 1990) 392396. 7. P. Monthoux, A. Balatsky, and D. Pines,
D. Pines / Spin fluctuations and I~igh temperature superc~nduct.ri~
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