nating the kinetics of water release and
mobility helping to reduce the energetic
costs of correlated, local, side-chain movements
(1). In effect, the proposed mechanisms
ofantigen-antibov union show striking
similarities to the induced fit mechanisms
often implied in enzyme-substrate
(15, 20) and DNA-protein interactions (24).
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2. D. C. Wiley, I. A. Wilson, J. J. Skehel, Nature
(London) 289, 373 (1981); H. A. Alexander et a.,
ibid. 306, 697 (1983).
3. A. G. Amit, R. A. Mariuzza, S. E. V. Phillips, R. J.
Poljak, Sacenc 233, 747 (1986).
4. D. C. Benjamin at al., Annu. Rev. Immundo. 2, 67
(1984).
5. T. M. Fieser, unpublished results.
6. B. Schecter, I. Schecter, J. Ramachandran, A. Conway-Jacobs,
M. Sela, Eur. J. Biocbem. 20, 301
(1971); B. Schecter, A. Conway-Jacobs, M. Sela,
ibid., p. 321; A. Yaron, E. Katchalski, A. Berger
Biopdymes 10, 1107 (1971); V. J. Hruby, in Pcrspectives
in Peptide Chemic9y, A. Eberkc, R. Geiger,
T. Weiland, Eds. (Karger, Basel, 1981), p. 207; Y.
Paterson, Biochemietry 24, 1048 (1985).
7. C. Chothia and J. Janin, Nature (London) 256, 705
(1975); E. D. Getzoff, J. A. Tainer, A. J. Olson,
Biophys.J. 49, 191 (1986).
8. J. A. Berzofsky, Scienc 229, 932 (1985), induding
the references therein.
9. L. M. Amzel, R. J. Poljak, F. Saul, J. M. Varga, F. F.
Richards, Proc. Nati. Acad. Sci. USA. 71, 1427
(1974); E. A. Padlan, D. R. Davies, S. Rudikoff, M.
Potter,Immunocbemmstr 13, 945 (1976); S. W. Suh
ct al., Proteins 1, 74 (1986).
10. P. M. Colman, J. N. Vargese, W. G. Laver, Nature
(London) 303, 41 (1983); D. W. Fannmiz, J. A.
Smith, G. D. Rose, Bi7oome 25, 863 (1%96).
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(1972); M. Lubeck andW. Gerhard, Virol~118, 1
(1982); A. G. Diamond, G. W. Butcher, J. C.
Howard,J. Immunot. 132, 1169 (1984).
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Karplus, Biopotyee 25, 1909 (1986).
13. Affinity measurements were performed by overnight
incubation of serially diluted '25I-MHr with antibodies
to MHr (anti-MHr) purified over Sepharosepeptide
columns. Purified MHr (100 Iqg) was labeed
with 125I by the chloramine T method and
dialyzed ovemight to remove unbound 125I. The
specific activity ofthe labeld MHr was calculated to
be 14.2 Ci/mmol on the basis ofprecipitation by 10
percent trichloroacetic acid. The MHr antisera were
affinity-purified over peptides conjugated to Sepharose
beads by cyanogen bromide. Purified antibies
were cluted by an abrupt change ofpH (diethylamine,pH
11.0), neutralized immediately, and concentrated
by dialysis ayainst polyethylene glycol.
After overnight incubation of serially diluted 125Ilabeled
MHr with purified anti-MHr, the antigenantibody
complexes were precipitated with 40 pA of
Staphylococcs aureus, washed once with phosphatebuffered
saline (10 w sodiumo hcosphate, O.15M
NaCl, pH 7.2) and twice with 500 mM LiCl and
100 mM tris, pH 8.0, and the radioactivity was
counted. Dilutions of MHr ranged from 3 x 10-'
to 3 x 10-9M. Each concentratnon point was done
in triplicate and the background values for the
"preimmune" serum were subtracted for each point.
Affinity constants were derived by Scatchard analysis,
with the ratio of the bound to the unbound
plotted as a function of the bound, and were expressed
as dissociation constants [J. A. Bcrzofsky
and I. J. Berkower, in FundamentalImmunotog,W.
E. Paul, Ed. (Raven, New York, 1984), p. 600].
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Lintan and R. A. Good, Eds. (Plenum, New York,
1978), p. 85.
15. T. E. Creighton, Proteins: Structures and Moecutar
Propcries (Freeman, New York, 1983), pp. 299,
426.
16. F. A. Rkhards, Rev. Biophys. Bio . 6, 151 (1977).
17. A. B. Edmundson and K. R. Ely, in Synthetic
Peptides asAntgens, Ciba Sy. 119, R. Porter and
J. Whelan, Eds. (Wilcy, New York, 1986), p. 107.
18. J. A. Tainer, E. D. Getzoff, Y. Paterson, A. J. Olson,
R. A.Lemer,Annu. Rep. Immuno. 3, 501 (1985).
19. A. R. Fersht et al., Nature (London) 314, 235
(1985); R. J. Leatherbarrow and A. R Fersht,
Protein Eng. 1, 7 (1986).
20. D. E. Koshland, Jr., in The Enzymes, P. D. Boyer,
Ed. (Academic Press, New York, 1970), voL I p
341; W. P. Jencks, Adv. Enzymol. 43, 219 (1975-)
D. E. Koshland, FEBS Lett. 62, E47-E52 (1976);
A. Fersht, Enzyme Strucure and Meccanism (Freeman,
New York, 1985), pp. 25, 311.
21. C. Chothia, A. M. Lesk, G. G. Dodson, D. C.
Hodgkin, Nature (London) 302, 500 (1983); G. D.
Smith, at al., Proc. Nat. Acad. Sci. USA. 81, 7093
(1984).
22. E. E. Howell, J. E. Villafranca, M. S. Warren, S. J.
Oatley, J. Kraut, Scince231, 1123 (1986); R. Bott
at al., Am. Chem. Soc. Symp. Ser., in press.
23. E. Westhof at al., Nature (London) 311, 123
(1984); J. A. Tainer at al., ibid. 312, 127 (1984).
R. J. P. Williams and G. R. Moore, Trends Biochm.
Sci. 10 (1985); J. Novotn~a al., Proc. Natl. Acad.
Sci. U.SA. 83, 226 (1986); J. M. Thornton, M. S.
Edwards, W. R. Taylor, D. J. Barlow, EMBOJ. 5,
409 (1986).
24. J. A. McClarin at al., Sciecew 234, 1526 (1986).
RCECENTLY HIGH-TEMPERATURE SUperconductivity
has been observed
in a number of doped lanthanum
copper oxides near a metal-insulator transition
(1), a pattern exhibited previously by
(Ba,Pb)BiO3 (2). The crystal structure suggests
that the Cu2' is in an S = 1/2, orbitally
nondegenerate state, strongly hybridized
with the surrounding oxygen p-levels, and
this is in agreement with high-temperature
magnetic data (3) on the stoichiometric,
insulating compound La2CuO4.
The appropriate model seems to be the
basic nearly half-filled Hubbard model (4)
with moderately large repulsion energy U
and antiferromagnetic exchange constant
I = t2/U where t is the site-hopping matrix
element. The K2NiF4 structure is a wellknown
case in which the magnetic layers are
relatively weakly interacting, and in the temperature
range 30 to 70 K we can assume
magnetic two-dimensionality. This led me
Departnent ofPhysics, Princeton University, Princeton,
NJ 08544.
25. H. M. Geysen, R. H. Meloen, S. J. Barteling, Proc.
Nati. Acad. Sci. USA. 81, 3998 (1984); H. M.
Geysen, S. J. Barteling, R. H. Meloen, ibid. 82, 178
(1985); S. J. Rodda, H. M. Geysen, T. J. Mason, P.
G. Schoofs, Md. Immunod. 23, 603 (1986).
26. W. A. Hendrickson, G. L. Klippenstein, K. B.
Ward, Proc. Nad. Acad. Sci. USA. 72, 2160
(1975); S. Sheriff, W. A. Hendrickson, R. E. Stenkamp,
L. C. Sieker, L. H. Jensen, ibid. 82, 1104
(1985); S. Sherff, W. A. Hendrickson, J. A. Smith,
J. Md. Bio., im press.
27. M. L. Connolly, Scienc 221, 709 (1983); J. Mo.
Graphic 3, 19 (1985).
28. E. D. Gtzoffand J. A. Tainer, unpublished method.
29. We thank S. Sheriff and W. A. hendrickson for use
of their MHr coordinates and temperature factors
corrected for crystal contacts; R. Bott, D. R. Davics,
W. A. Hendrickson, W. P. Jcncks, N. Klinman, A. J.
Olson, E. Padlan, Y. Paterson, M. E. Pique, J. Sayre,
S. Sheriff, and A. Tramontano, among others, for
helpful discussions regarding the manuscript. Supported
in part by grants from the National Institutes
of Health (E.D.GUand R.A.L.).
11 July 1986; accepted 31 December 1986
to reexamine the idea of the "resonating
valence-bond" (RVB) state (5).
Early doubts about the nature of the
ground state of the antiferromagnetic Heisenberg
Hamiltonian
= J2si (1)
of Hulthen (6) and Marshall (7) (where ij
is the spin at site i and innjindicates summation
over nearest neighbors i andj) seemed
to have been laid to rest by arguments from
quantum fluctuations of spin waves in the
NMl state (8) in >1 cdimension, and by
experimental observations of antiferromagnetism.
In 1973, however, Anderson (5)
proposed that, at least in the triangular twodimensional
antiferromagnet for S = 1/2,
and perhaps in other cases, the ground state
might be the analog of the precise singlet in
the Bethe solution of the linear antiferromagnetic
chain (6). In both cases, the zeroth
order energy of a state consisting purely of
nearest neighbor singlet pairs is more nearly
realistic than that of the Neel state, and I
SCIENCE, VOL. 235
The Resonating Valence Bond State in La2CuO4
and Superconductivity
P. W. ANDERSON
The oxide superconductors, particularly those recently discovered that are based on
La2CuO4, have a set ofpeculiarities that suggest a common, unique mehanism: they
tend in every case to occur near a metal-insulator transition into an odd-electron
insulator with peculiar magnetic properties. This insulating phase is proposed to be the
long-sought "resonating-valence-bond" state or "quantum spin liquid" hypothesized
in 1973. This insulating magnetic phase is favored by low spin, low dimensionality,
and magnetic frustration. The preeistig magnetic singlet pairs ofthe insulating state
become charged superconducting pairs when the insulator is doped sufficiently
strongly. The mechanism for superconductivity is hence predominantly electronic and
magnetic, although weak phonon interactions may favor the state. Many unusual
properties are predicted, especiafly of the insulating state.
1196
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proposed that higher order corrections, allowing
the singlet pairs to move or "resonate"
a la Pauling, might make this insulating
singlet or RVB state more stable. This
state is quite dearly distinct from two other
locally stable possibilities, the NMl state and
the 'spin-Peierls" state of a self-trapped,
localized array ofsinglet pairs. Each ofthese
other states has a broken symmetry relative
to the high-temperature paramagnetic state,
and would exhibit a phase transition with
temperature. Fazekas and Anderson (9) improved
on my numerical stability estimates,
and Hirakawa et al. (10) have proposed
application of the RVB state to specific
compounds.
The triangular lattice is not the only possible
candidate: the two-dimensional square
lattice, for instance, will undoubtedly exhibit
an RVB state if either or both of two
possibilities occurs: a next nearest neighbor
antiferromagnetic interaction strong
enough to frustrate the Neel state, or virtual
phonon interactions short of being strong
enough to allow a spin-Peierls instability.
[Hirsch (11) has shown numerically that the
simple square lattice probably retains a magnetization
but that finite U may favor the
RVB state.] It is our hypothesis that pure
La2CuO4 is in an RVB state; this proposal is
supported to some extent by the magnetic
susceptibility data ofGanguly and Rao (3).
It is not easy to calculate with RVB states
or to represent them. I want to give here a
representation in terms of Gutzwiller-type
projections ofmobile-electron states, which
is probably not particularly useful computationally
but is suggestive. This representation
was in turn suggested by Rice and
Joynt's remarks (12) on the Gutzwiller approximation
to the Bethe solution.
A single pair of electrons in a mobile
valence bond along a lattice vector T may be
written
b+r to==
(xCj Cj+l)l =0
(2)
where b+ is the electron-pair creation operator,
cj+ is the single-electron creation operator,
and N is the total number of sites. A
linear combination of all nearest neighbor
bonds may be written
bnn= LbT+
T=(XX)
where nn indicates summation over nearest
neighbors, or in general, ifwe want a distribution
of bond lengths, the linear combination
may be written as
b = Ia(k) cktT c+A4 (3)
A
E
N n
B _
n N
Fig. 1. Total energy E as a function of average
occupation number n. (A) Insulating case. (B,
Metallic case. N is the total number oflattice sites.
with the condition on the expansion coeffi-
cients
2 a(k) = 0
k
(4)
ifwe do not allow double occupancy.
Unfortunately, if we now try to make a
Bose condensation ofN electrons in mobile
valence bond states by forming
= (b+)N/2 to (5)
this state contains large numbers of empty
and doubly occupied sites. To make up a
genuine RVB state we may try one of two
roughly equivalent projection techniques.
The simplest is the straightforward Gutzwiller
method: form the projection operator
Pd =U ( - ntnil) (6)
where ni is the occupation number, and
'RVB = Pd(b )N2'Po (7)
is our trial wave function.
We may also use quasifermion operators
with the double occupancy projected out
(13), for example,
= c4 (1 - n)
and write
b = a(k)c4kt c-A (8)
k
in Eq. 5.
At this point we resort to the Dyson
transformation between product states and
Bardeen-Cooper-Schrieffer (BCS) states. As
shown by Dyson (14), if we note that
(b,k)2 = 0 (with bkT = Ck1 CA4),
1 + akbk = exp(akbk)
hence
PBCS = fI(V'1- hk + Vbkbk)*0
exp [
k
kb+]'PO (9)
6 MARCH I987
where hk is the variational parameter in the
standard BCS treatment.
Ifwe project TBCS on the state with just
N12 pairs, we obtain
PN/2'PBCS = [ [V
k
]bkJ
or, in other words, Eq. 5 is just a projected
BCS function. These transformations are
not singular as they are for conventional
BCS. Thus our approximate TRVB is also
related to the appropriate projection of a
BCS function:
I
A1RVB = PN/2PdH[vvTN
ak 4t+%
+ V1+aaCk C
0 (11)
[1or =PN/iS2J [1a
+ ak C ]
V1+ k
In the insulating state every site i is filled
once so that lakl might as well be constant,
ak = + 1, and the wave function contains a
"pseudo-Fermi surface" at which ak changes
sign. There is no reason why this surface
should coincide with the Fermi surface of
any energy in the system, since the pseudoFermi
surface is controlled by the condition
that it divide k-space into equal halves and
by the choice of T values for the valence
bonds. The existence of a pseudo-Fermi
surface, I believe, is real and the spin excitations
may resemble those of a real Fermi
liquid. This would explain the experimental
observation of a Fermi-like susceptibility.
On the other hand, considered as a solution
to the Hubbard model, there is a gap
for any charged excitation. By assumption,
U is so large that adding the (N + 1)th
electron costs an extra -Uin energy, relative
to adding theNth, and the Fermi energy lies
in this gap for the stoichiometric compound.
There is a cusp in total energy as a
function of occupation number at n = N
(Fig. 1A). In this case the PN/2 projection
operator may not be omitted. One may
represent PNI2 explicitly by giving ak a phase
e' and writing
TRVB = Pdf doe-iN120
Ts I H'+ ]
Thus in this insulating state there is no
meaning to the phase ofak; a wave function
with fixed phase contains states ofnecessarily
widely different energy.
Now let us consider the state obtained if
we dope the system in order to remove the
REPORTS II97
/
Ck t C-A4 expi (k' T) APOk
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"half-filled" criterion and make it into a
metal. From a "mean field" point ofview, as
soon as the system is metallized it becomes a
superconductor, since the pairing already
exists in the RVB state, and an energy -J is
required to break a valence-bonded pair. As
shown in Fig. IB, as soon as the occupancy
leaves N, there is no cusp in the energy, the
compressibility becomes finite, and by the
standard arguments the state can acquire a
fixed 0 rather than n.
As a practical matter, the effective mass of
quasiparticles will be of order m* = WB,
where 8 is the fractional doping n =
N (1 - 8). Correspondingly, the coherence
length to will be oforder
kFto = EF 1
where kF and EF are the Fermi wave vector
and energy, respectively, A is the energy
gap, and the kinetic energy is of order tb.
Thus the transition temperature TF will at
first be dominated by phase fluctuations. (In
actual physical fact, at first the dopant ions
will be screened out by bound quasiparticles,
and it will take a finite dopant concentration
to metallize the sample.) The maximum
Tc, oforder or less than t2/U = J, will
occur when t/U -8 and kinetic and pairbinding
energies match. Pressure will increase
t and, within limits, increase T, as
well.
From a theoretical point ofview, the most
exotic feature of these experimental results
(1) is that they confirm the existence of a
new liquid, only conjectured previously (3,
5, 10). This liquid is insulating only by
virtue of a "commensurability gap," and
therefore resembles the Laughlin state in the
fractional quantum Hall effect (15). Both of
these states may be described as "Mott liquids"
since the basic physics is that of the
"Mott transition" (16), and their key feature
is that there is no symmetry breaking vis-avis
the high-temperature state.
There are several experimental consequences
of the above. The key point is the
observation of the RVB state in the stoichiometric
La2CuO4, which should be easy
with neutrons, especially ifthere is a pseudoFermi
surface. Second, the pseudo-Fermi
surface may or may not cross the real Fermi
surface, defining lines of zeroes of the gap
function. The occurrence of lines of zeroes
and of antiferromagnetic correlations in
some heavy fermion superconductors suggests
a family resemblance to the RVB state,
although the parameter values are totally
different.
Finally, I would call attention to the
numerous unreproducible reports of hightemperature
superconductivity in special
samples of CuCl. In every case it is reason-
II98
able to imagine a surface layer ofCu2+ with
or without the appropriate degree ofoxidation;
such reports should sharpen the search
for still more RVB superconductors. It is
also noteworthy that the first oxide superconductor,
Li2TiO4 (17), dosely resembles
NaTiO2, the only other likely RVB material.
REFERENCES AND NOTES
1. J. G. Bednorz and K. A.Miller, Z. Phys. B 64, 189
(1986); J. G. Bednorz, M. Takashige, K. A. Muller,
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7. W. Marshall, Proc. R. Soc. London Ser. A 232, 48
(1955).
of ocean.
IDAL CURRENTS FLOWING OFF THE
continental shelf or across reefs or
banks produce large internal waves
(1, 2). Those waves formed at the shelf
break propagate onshore (3). Surface currents
over the waves produce alternating
zones of convergence and divergence. Flotsam
swept into the convergence, if buoyant
enough, will be trapped there and carried
onshore. Some types of larval invertebrates
and fish migrate onshore by this mechanism
(4, 5). Most oils float on water and potentially
could be carried onshore by internal
waves. This study shows that floating oil
(spilled asphalt) was swept up by currents
over a set ofinternal waves, concentrated in
the convergence zones over the waves, and
transported onshore. To my knowledge, this
is the first time that onshore transport of an
oil spill by internal waves has been observed
8. P. W. Anderson, Phys. Rev. 86, 694 (1952).
9. P. Fazekas and P. W. Anderson, Philos. Ag. 30,
432 (1974).
10. K. Hirakawa, H. Kadowaki, K. Ubikoshi, J. Phys.
Soc.Jpn. 54, 3526 (1985); I. Yamada, K. Ubikoshi,
K. Hirakawa, ibid., p. 3571.
11. J. E. Hirsch, Phys. Rev. B 31, 4403 (1985); Phys.
Rev. Lest. 54, 1317 (1985). Hirsch's simulations in
these references are very sugestive confirmation of
the RVB model of superconductivity.
12. T. M. Rice and R. Joynt, paper presented at the
International Conference on Valence Fluctuations,
Bangalore, 1987.
13. I am indebted for this suggestion to B. S. Shastry
and T. V. Ramakrishnan.
14. F. J. Dyson, personal communication.
15. R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
16. N. F. Mott, Proc. Phys. Soc. London Scct. A 62, 416
(1949).
17. D. C. Johnson, H. Prakash, W. H. Zachariasen, R.
Viswanathan, Mater. Res. BuU. 8, 777 (1973).
18. I would like to acknowledge the hospitability ofthe
Council for Scientific and Industrial Research and
the Tata Institute at Bombay, in hosting my visit to
Bangalore, where the key parts of this work were
done. Discussions with many people there were
helpful, especially T. V. Ramakrishnan, B. S.
Shastry, C. M. Varma, J. Hirsch, and T. M. Rice.
Also I would like to acknowledge discussions with I.
Affleck and T. H. Geballe, and that my faith in the
RVB state was sustained for many years by P.
Fazekas. The manuscnrpt was prepared while P.W.A.
was a Fairchild Scholar at the Califomia Institute of
Technology. This work was partially supported by
NSF grant DMR 851-8163.
23 January 1987; accepted 3 February 1987
and recognized, although it might be expected
from the known properties of internal
waves (6).
On 21 June 1985 a large (about 1.5 x
106 liters) tank ofliquid asphalt burst at the
Morehead City port in North Carolina
(7505'N, 3504'W) (Fig. 1). Some of this
asphalt spilled into the Beaufort Inlet and
was carried out to sea by the tide.
Three days later surface plankton samples
(from the top 26 cm of the water column)
were collected with a Manta net (7) (26 by
96 cm, mouth opening; 0.333 mm, net
mesh) from the waters around a set of
internal waves (8). The set ofinternal waves
was about 4 km offshore (Fig. 1). Replicate
(n = 3) 5-minute tows (>50 m3 of water
filtered) were made in front of and behind
Institute ofMarine Sciences, University ofNorth Carolina
at Chapel Hill, Morehead City, NC 28557.
SCIENCE, VOL. 235
The Onshore Transport of an Oil Spill by
Internal Waves
ALAN L. SHANKS
Internal waves generated by tidal currents concentrated and transported an oil spill
(liquid asphalt) onshore. Plankton net samples were collected in front ofand behind a
set ofinternal waves as well as in the convergence and divergence zones over the waves.
Tar "balls" were most abundant (greater than 30-fold) in the samples from the
convergence zone. Comparison ofthe abundance oftar balls in front ofand behind the
set of waves suggests that the internal waves "caught" about 68% of the asphalt
encountered and concentrated and swept shoreward tar balls from almost 8 kilometers
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and Superconductivity4CuO2The Resonating Valence Bond State in La
P. W. ANDERSON
DOI: 10.1126/science.235.4793.1196
(4793), 1196-1198.235Science
ARTICLE TOOLS http://science.sciencemag.org/content/235/4793/1196
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