Present status of the theory of the high-Tc cuprates
The Gutzwiller projected mean-field theory, also called plain vanilla or renormalized meanfield theory, is explained and its successes and possible extensions in describing the phenomenology of the cuprate superconductors are discussed. Throughout, we emphasize that while this is a Hartree—Fock b...
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Anderson, P.W. 2017-06-11T07:39:34Z 2017-06-11T07:39:34Z 2006 Present status of the theory of the high-Tc cuprates / P.W. Anderson // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 381–390. — Бібліогр.: 32 назв. — англ. 0132-6414 PACS: 74.72.–h, 74.20.–z https://nasplib.isofts.kiev.ua/handle/123456789/120154 The Gutzwiller projected mean-field theory, also called plain vanilla or renormalized meanfield theory, is explained and its successes and possible extensions in describing the phenomenology of the cuprate superconductors are discussed. Throughout, we emphasize that while this is a Hartree—Fock based BCS theory, it embodies fundamental differences from conventional perturbative many body theory which may be characterized by calling it a theory of the doped Mott insulator. I have first of all to acknowledge my long-term collaborator Nai-Phuan Ong, for the innumerable times he has helped me with experimental know-how and theoretical comments. I should also acknowledge my collaborators in the plain vanilla exercise, T.M. Rice, P.A. Lee, Mohit Randeria, Nandini Trivedi and Fu-Chun Zhang, as well as others who were involved in finding the solution 17 years too early: Claudius Gros and Gabi Kotliar. Others who have helped keep my mind clear about experimental data have been Doug Bonn, Nicole Bontemps, Bernhard Keimer, Seamus Davis, Mike Norman, J.-C. Campuzano, Tom Timusk, Kam Moler; this is only a tiny fraction of the totality of individuals who have been helpful. But none of the above need take any responsibility for what I say here. Finally, there is my good friend and sounding board, V. Muthukumar. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур General Aspects Present status of the theory of the high-Tc cuprates Article published earlier |
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present status of the theory of the high-tc cuprates |
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Anderson, P.W. |
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Anderson, P.W. |
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General Aspects |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
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The Gutzwiller projected mean-field theory, also called plain vanilla or renormalized meanfield
theory, is explained and its successes and possible extensions in describing the phenomenology
of the cuprate superconductors are discussed. Throughout, we emphasize that while this
is a Hartree—Fock based BCS theory, it embodies fundamental differences from conventional
perturbative many body theory which may be characterized by calling it a theory of the doped
Mott insulator.
|
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0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/120154 |
| citation_txt |
Present status of the theory of the high-Tc cuprates / P.W. Anderson // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 381–390. — Бібліогр.: 32 назв. — англ. |
| work_keys_str_mv |
AT andersonpw presentstatusofthetheoryofthehightccuprates |
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2025-11-25T23:46:37Z |
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2025-11-25T23:46:37Z |
| _version_ |
1850583676380774400 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5, p. 381–390
Present status of the theory of the high-Tc cuprates
P.W. Anderson
Department of Physics, Princeton University, Princeton NJ 08544, USA
E-mail: pwa@princeton.edu
Received October 3, 2005
The Gutzwiller projected mean-field theory, also called plain vanilla or renormalized mean-
field theory, is explained and its successes and possible extensions in describing the pheno-
menology of the cuprate superconductors are discussed. Throughout, we emphasize that while this
is a Hartree—Fock based BCS theory, it embodies fundamental differences from conventional
perturbative many body theory which may be characterized by calling it a theory of the doped
Mott insulator.
PACS: 74.72.–h, 74.20.–z
Keywords: mean-field theory, high-Tc superconductivity.
1. Historical note
In early 1987, just as the remarkable Bed-
norz—Muller discovery was becoming widely known,
the basis for the theory of the materials which they
had discovered was laid down [1]. It was observed
that the CuO2 planes on which they are based were
plausibly describable by a particularly simple version
of the Hubbard model, the case of a single non-
degenerate band, and that the «stoichiometric» case
where the nominal valence is Cu� � is well described
as a Mott insulator. The superconductors are obtained
when, in the «reservoir» layers between the planes,
substitutional impurities of lower valence are intro-
duced, thus doping extra holes into the Cu d-shell
(which is of course strongly hybridized with the O
p-shells, according to the well-known principles of
ligand field theory). All of the plausible theories
about these materials describe them as «doped Mott
insulators».
A mechanism for electron pairing in mixed valence
systems, which are somewhat similar, already had
been suggested by two groups [2], namely using the
antiferromagnetic «superexchange» interaction bet-
ween spins as a pairing force. In Ref. 1, I likened this
pairing force to the valence bonding effect for which it
is essentially responsible, and pointed out that the old
idea of a quantum liquid of valence bonds resonating
around among different pairings of atoms had a great
similarity to superconductivity. In fact, I proposed an
explicit form for such a state in terms of a Gutz-
willer-projected BCS paired wave function, and in a
series of papers in 1987 elaborated on formalisms for
getting continuously from the Mott insulator to the
superconductor.
Unfortunately, through a series of misjudgments
on my part, which are permanently recorded in an
unfortunately timed book [3], my group and I there-
upon fell off the correct trail to a solution, only to
return to the correct path ten years later once we had
absorbed the unequivocal experimental evidence that
my «interlayer» theory was wrong. But fortunately,
at least two separate groups had in the meantime built
a theory on the 1987 foundations which turned out to
be basically correct [4,5]. In this article I will follow
the second of these references but they are equivalent.
The important thing about both is that they realized
that the correct solution of the original undoped
resonating valence bond (RVB) problem was not the
isotropic «extended s» which I had been discussing
but a more complex one with both s-like and d-like
gaps, which Kotliar called «s id� ». Both of these
papers predicted the real d-wave gap with nodes which
was eventually observed, and in addition a number of
other results which were to be confirmed one by one in
the coming years. It has been our perverse fate that the
theory, properly handled, has made one after another
correct prediction, well ahead of the experiments, but
that these have been obscured by irrelevancies and
© P.W. Anderson, 2006
misinterpretations until the mistaken impression has
arisen that the whole subject is utterly mysterious.
It was not for another 5 years that the d-wave gap
was verified, and by that time the field had suffered
from a proliferation of proposed theories of greater or
lesser degrees of plausibility. The gradual experi-
mental unveiling of the facts about the cuprates
sometimes meant that each experiment came with a
built-in theory and that theories which had predicted
the result long before were not sufficiently «up-to-
date» to enter the public discussion. For instance,
the d-wave came to be identified with the idea of
propagating «antiferromagnetic spin fluctuations»,
which was a popular fad at the time of its verification,
rather than with its earliest, and much more natural,
prediction in Refs. 4, 5. Another example of this
phenomenon was the observation of the «spin gap» or
«pseudogap» in underdoped materials above the su-
perconducting «dome», again an obvious consequence
in Refs. 4, 5, but as it revealed itself it received a
congeries of faddish explanations from local theorists:
a mysterious «quantum critical point», a «spin ne-
matic», again AF spin fluctuations, the «d-density
wave», you name it.
In any case, these early theories only came to be
revived in the early 2000's by groups which were able
to use them as the basis for accurate quantum Monte
Carlo calculations using realistic parameter values
[6,7], and brought forward without too much mo-
dification some of the predictions which had looked so
surprising in 1988 but had been very close to correct.
A group of us summarized the successes of the theory,
adding a small amount of further physical ideas, in a
review paper which we called the «plain vanilla»
theory of high Tc [8]. Here I will review that theory
and the subsequent developments, including parti-
cularly the explanation and calculation of asymmetric
tunneling spectra using it, and the recent theory of the
pseudogap phase which throws a great deal of light on
the overall physics of the phase diagram.
Since a great deal of emphasize has been put on the
problem of the epistemics of complex phases like the
high-Tc cuprates, and whether a meaningful solution
to the accompanying puzzles can be found, I'd like to
spend a few sentences on that aspect. First, a bit about
the nature of condensed matter physics. Among the
sciences this one is almost uniquely overdetermined,
experimentally because of the variety and precision of
the probes which can be applied, and theoretically
because the quantum physics of atoms and electrons is
so well understood. I have always maintained that the
correctness of a theoretical hypothesis is assured in
this field if it can find a way to fit in with all these
constraints: that there is likely to be only one possible
way to fit all— or even a majority — of the obser-
vations together, and not to violate any theoretical
impossibilities. In this process of fitting things to-
gether there is no room for one-experiment theories,
doctrinal conservatism (the older generation and some
younger scientists won't let go of phonons), or yet un-
trammeled imagination (anyon superconductivity,
SO(5), QCP’s, perhaps interlayer tunneling). The
naked reality is strange enough.
A final word. The way you know you are right is
when you wake up and realize that you have the
answers to deep, fundamental questions that you
didn't really know to ask or expect to answer. For the
old superconductors, such a question was «why are
polyelectronic metals favored?» — the question Pi-
nes, Morel, myself and McMillan answered with
dynamic screening for the phonon theory. Here there
are at least two such questions: «Why the cuprates —
what is unique about copper?»; and, «Why d-wave
and why is the gap persistently real? That is, why the
striking nodes?» The second is the question I didn't
think to ask, but it is profound — any other simple
mechanism which leads to a d-wave can lead also to an
xy or isotropic symmetry, which will appear in
quadrature in order to fill in the nodes, which are
intrinsically unstable in a BCS theory. The mechanism
by which the A phase of 3He acquires nodes was, for
instance, crucial to our understanding of that system.
2. The plain vanilla (RMFT) theory
The underlying concept of the plain vanilla theory
is very simple. In fact, it follows as closely as possible
the precedent of the BCS theory. The BCS theory in
its original form is a generalization of Hartree—Fock
theory to allow for not only the direct and exchange
mean fields, which appear in the one-electron mean
field Hamiltonian as V r r v r r( ) ( ) ( ) ( )*� � � � , and
A r r r r( , ) ( ) ( )*� �� � , but also the «anomalous» self-
energy, � � �( , ) ( ) ( )* *r r r r� � � h.c. These result from
the three possible ways to factorize the interaction
energy,
V r r r r r r drdr( ) ( ) ( ) ( ) ( )* *� � � � � �� � � � .
BCS theory is basically a variational theory: the
assumed wave function is a simple product of one-
quasiparticle operators creating quasiparticles from
the vacuum, and the «gap» equations, equivalently to
the mean-field equations, determine that the quasi-
particle creation operators all have positive energies,
so that all possible single-particle excitations increase
the energy.
Simple Hartree theory won't work for a Hubbard
model in which the on-site interaction energy is the
382 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
P.W. Anderson
largest energy in the problem. Very early on [9], it
was realized that the solution to that problem was to
transform to a representation in which the on-site
interaction energy U has been renormalized to � as
opposed to the conventional scheme well described by
Shankar [10] where the idea is to transform to some
system of noninteracting entities. We employ a ca-
nonical transform exp ( )iS to eliminate all matrix
elements of the Hamiltonian which lead into the
subspace in which two electrons simultaneously oc-
cupy the same site; i.e., those which have the large
energy U. This transformation [9] can be derived
perturbatively as a series in inverse powers ofU. That
is, we start from the «real» Hamiltonian
H t c c U n nij
ij
i j i
i
i0 � � �
� �
� �� �
�
� �
† h. c. , (1)
along with direct exchange and smaller terms, and
transform it into the t J� Hamiltonian;
H H H O t UiS iS
t J0 0
3 2 � � ��
�e e ( ) .../ , where
Ht J� is given by,
H t P c c Pt J ij
ij
G i j G�
� �
� � ��
�
� �
† h. c.
�
�
� �
�J n nij
ij
i j i j( )S S
1
4
. (2)
In the above equation, we have ignored terms in-
cluding longer-range Coulomb and phonon interac-
tions — which latter are not particularly small, but
clearly are incapable of causing the gigantic super-
conducting gaps which are observed. Here, PG is
the full Gutzwiller projector which hereafter we
will call P:
P n ni
i
i� � � ��( )1 . (3)
That (2) is really a correct description of the
electronics of the cuprates was tested first by Schluter
et al. [11], in 1988, who found that the calculated
energies of low-lying states in small clusters of the
cuprate structure, using the full Hamiltonian, were
well reproduced by the truncations implicit in equa-
tions (1) and (2). (Another early discovery long since
forgotten.)
Always remembering that the t J� model (Eq. (2))
wave function must be transformed by exp ( )iS in the
end to represent Hubbard model reality, we proceed
to try to find a variational ground state for (2).
Clearly, since the Hamiltonian is now in block
diagonal form, any low-energy state must contain only
amplitudes for the projected subspace, so that,
� �� P r r rN( , , , )1 2 � , (4)
where � is a general N-particle wave function. The
essence of the «plain vanilla» approximation is to
propose that we approximate �, the wave function to
be projected, using the Hartree-Fock-BCS ansatz that
it is a product of quasiparticles. I can see no reason
that this is apriori less reasonable than the BCS
theory itself. If there is a single-particle-like repre-
sentation of the ground state, this is the way to derive
one. In the event, there is such a representation,
experimentally — by now there is all kinds of evi-
dence that the state has gapped quasiparticles near a
large Fermi surface, over a fairly wide range of
doping — the so-called «dome» region of the phase
diagram of T vs doping. I can't too much emphasize
this: this procedure is the natural, and probably the
only, way to derive a BCS-like superconductor from
the t J� Hamiltonian.
A second, and less certain, fact is that the resulting
excitations may be reasonably sharp and well-defin-
ed-though, because of the projection operator, the
same may not be said of actual quasiparticles: cP is not
the same as the single-particle-like excitation Pc. But
the representation in terms of Pc's has some as yet
unresolved peculiarities: it is overcomplete, which
may mean, among other things, that the excitations
can scatter each other very strongly. But the fact of
overcompleteness does not much affect either the
variational equations nor the validity of them as
giving the energies of approximate single-particle
excitations. In writing out these equations we follow
Ref. 5 in self-consistently choosing a particular rela-
tive gauge [12] for the J-term relative to the kinetic
energy. This choice is discussed later.
Our ansatz for � in Eq. (4), then, is
� � � �� � � �
( )|† †u v c ck
k
k k k
0 . (5)
In the recent papers by Paramekanti et al., the
parameters u and v were evaluated variationally using
variational Monte Carlo techniques [6]. But the
results were almost identical to those found in the
earlier papers using a very simple approximation due
to Gutzwiller, which is exact in the limit that the gap
is small relative to the Fermi energy. In this ap-
proximation we assume that the correction to the
probability of occupation of the sites caused by pro-
jection is uncorrelated spatially, because, obviously,
the projection operates only site by site, ignoring the
occupancy of neighbors. Thus the correction may be
estimated by simply calculating what happens to the
average occupancies. It is easily shown that the
change in the average number of neighbors with one
site empty, the other singly-occupied, is a reduction
by the factor g x x� �2 1/( ), while the change in the
Present status of the theory of the high-Tc cuprates
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 383
number of pairs of singly-occupied sites is an increase
by g x gJ � � � �4 1 22 2/( ) ( ) . Thus the effect of the
kinetic energy is reduced by the factor g, and that of
J is increased by gJ, but otherwise, in this approxi-
mation, we employ the t J� Hamiltonian (2) in
precisely the same way as a real one. Thus we arrive
at the «plain vanilla» gap equations in the «Gutz-
willer approximation», i.e., the renormalized mean
field theory (RMFT):
�
�
k J k k
k
k
k
g J
E
� � �
�
�
�
�
2
, Ek k k
2 2 2� �� � ,
� � �
�
k k k k J k k
k
k
k
g g g J
E
� � � � � � �
�
�
�
� 2
.
(6)
Here J k k
� � is the Fourier transform of the exchange
interaction (assumed nearest neighbor), �k is the
bare, unrenormalized kinetic energy, � and � are the
anomalous and normal self energies, � is the renorma-
lized kinetic energy and Ek is the quasiparticle
energy.
In Fig. 1, we present results for the magnitude of
the d-wave gap, �, and the size of the order parameter
from Edegger et al.'s solutions of the gap equations
[13], just to convince the reader that these track the
observed maximum gap and dome reasonably well. A
generalized phase diagram incorporating the results of
a number of experiments is shown in Fig. 2. (This
figure differs from a phase diagram often drawn for
which the T* line intersects the dome and no trace of
the pseudogap phase remains for optimally doped
materials. Ong’s Nernst effect data among others seem
to unequivocally reject this interpretation.) Since
1988, it seems, the quantitative explanation of high-Tc
superconductivity has been available.
3. Extensions of the RMFT
3.1. Spin-charge locking
Note that as g 0 (the «true» RVB), � and � are
interchangeable. This represents a deep reality: that
for the half-filled Mott insulator, the representation
of the magnetic state of the spins by fermionic vari-
ables — the «spinons» of RVB theory — is doubly
overcomplete. One may represent an � spin on site i
either by creating an � spin, c
i�
† , or by destroying a �
on that site, ci� , or by any unitary superposition of
the two. In terms of a hypothecated RVB state,
described as a Gutzwiller projected BCS wave func-
tion at half filling, this means that the three An-
derson—Nambu spinors � i (i � 1 2 3, , ) of the BCS
state may be rotated at will, since they represent
quantities which transform into each other when the
SU(2) transformation is applied. The constraint of the
Gutzwiller projection also requires that only two of
the three � vectors have finite self-energies attached to
them, so that the symmetry is fully expressed as local
rotation of a dyad of self-energies � and � which must
be perpendicular to each other. All of the various
alternative states which have been proposed — the
«flux phase», the d-density wave, the staggered flux
phase, etc., are one or another of these totally
equivalent states, in the half-filled case. The two
«gaps», for the minimum-energy solution of Eq. (6),
are of maximally different symmetries. In the half-
filled case, and in the special case that we have only
nearest neighbor exchange so that
is of the form
cos cosk kx y� , the two are equal in magnitude and of
the form cos cosk kx y� . The only point where both
vanish is where both k's are � /2, which gives the
nodes which are the common feature of all the equi-
valent «ghost states» I mentioned above.
It is irrelevant that the actual half-filled band is
not the RVB state but a commensurate antiferro-
magnet, which has slightly lower energy for the
Heisenberg model. It is still meaningful to examine
the solutions of the full gap equation by referring
them back to the hypothetical limit g � 0. What
happens is that, as we reintroduce the kinetic energy
by doping, the antiferromagnetic state does not gain
kinetic energy as rapidly as the best RVB state, and
384 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
P.W. Anderson
0.1 0.2 0.3 0.4
x
0
0.1
0.2
0.3
0.4
0.2
0.4
0.6
�
k=
(
,0
)
(t
)
0 0.1 0.2 0.3 0.4
x
0
0.04
0.08
�
�
�
�'a b
Fig. 1. (a) Doping dependence of the dimensionless
mean-field parameters �, ��, �. (b) Doping dependence of
(solid) the SC order parameter, �, and (dashed) the gap,
| |�k , at k � ( , )�0 in units of t.
T, E
T, E
NFL
FL?
SC
x
??
Fig. 2. Generalized phase diagram of the cuprate super-
conductors.
the latter prevails at a few percent doping. Actually,
the equations (6) represent a special choice of gauge,
and we could in principle orient the kinetic energy
along any chosen axis in the �-space, and minimize the
energy as a functional of that orientation — the
resulting equations are given elsewhere [12]. But it is
clear that the optimum kinetic energy is achieved
when the «�» axis, the function with the symmetry
cos cosk kx y� , is chosen as an ordinary self-energy as
in (6). Then the other form of solution, the odd
combination cos cosk kx y� , acts in the direction �1
and serves as an anomalous self-energy or gap func-
tion. This is the principle I called «charge-spin lock-
ing» [12]. The locking energy was estimated in that
reference as well as by Kotliar and Liu and found to be
large: of order gt for small dopings and comparable
with T* for larger ones.
This large locking energy means that the gap
structure is established at temperatures well above the
superconducting «dome» of Tc 's. The reason the sys-
tem does not become superconducting is that the phase
stiffness is weaker, at least for doping up to the
optimum, than the gap energy, in contrast to the BCS
case. Tc is determined by the proliferation of vortices,
not by the breakdown of pairing. Experimentally, in
systems which are basically two-dimensional, one sees
Kosterlitz—Thouless transitions; and the cleanest
measurements for optimal YBCO find 3D XY model
exponents, very accurately [15]. Both observations
indicate that the order parameter amplitude remains
finite above Tc, and in fact the observations of Ong of
Nernst effect and nonlinear diamagnetic susceptibility
[16] show that a vortex liquid state persists well
above the dome, especially on the underdoped side.
From these measurements, as well as theory, we are
beginning to establish that what has been called the
«vortex liquid», i.e., a disordered superconductor as
opposed to a normal metal, may be a distinct state of
matter which is particularly characteristic of the
cuprates.
That Tc embodies a transition to a vortex liquid
state suggests a phenomenology of this metallic state
above Tc quite different from that of a normal metal.
We must think of it as everywhere superconducting,
but filled with a tangle of thermally-generated vor-
tices (at low fields). The supercurrent is fluctuating
arbitrarily and the state is characterized by a persis-
tence time � for the supercurrents: � � �J J t( ) ( )0
� � � �J t2 exp ( )/ � . One may estimate that � is self-
generated by the vortices themselves and is of order
h h n mV/ /� � 2 . The conductivity of such a vortex
tangle will be � �� ST. We may speculate that when
h/ � drops below kT, or equivalently when the number
of vortices drops below a critical value where their
entropy no longer compensates for their kinetic ener-
gy, the vortices evaporate: this is Tc, described in a
Kosterlitz—Thouless way as suggested by Lee [17].
This provides a basis for the empirical rule proposed
by Homes [18], as well as for the observations of
Timusk on anomalous increase of � in the pseudogap
region [19]. An even more speculative argument based
on the vortex tangle can explain the Nernst obser-
vations. (Fig. 3 shows a heuristic first attempt at a
description of the Nernst observations.) The fact that
Tc is controlled by the vortex liquid transition in-
validates most intuitions about it from BCS theory —
for instance, it makes the d-wave Tc insensitive to scat-
tering.
It is from the locking principle that the two
insights mentioned in the Introduction arise. Why the
CuO2 planes? Because they have the feature that
nearest-neighbor exchange with only four neighbors
allows the two almost degenerate gap functions of
even and odd symmetry in x and y, of which one may
be used to enhance the kinetic energy, the remaining
one giving a strong x y2 2� pairing energy. Of course,
there are other aspects, particularly the Jahn—Teller
distortion which enhances the energy scale, and the
fact that Cu� � � does not self-trap, and all mean that
unfortunately the scenario is unlikely to be repeated.
Why the nodes? Because the RVB can only be a dyad:
the spin interaction does not have a third possibility
for pairing. Thus only one function can be left over as
a gap function, and it must have nodal lines which do
not lie along the Fermi surface.
3.2. Hole-particle asymmetry
One of the more significant experimental anomalies
of the cuprates is the marked hole-particle asymmetry
of the vacuum tunneling spectra. To those of us who
worked on BCS superconductivity theory, this is
particularly striking because it is never observed in
Present status of the theory of the high-Tc cuprates
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 385
0 0.2 0.4 0.6 0.8 1.0
0.01
0.02
0.03
0.04
0.05
0.06
B, arb. units
T
h
e
rm
o
-e
m
f,
a
rb
. u
n
its
Fig. 3. A model calculation for the Nernst observations.
those materials. There is a large «peak-dip-hump»
structure observed on the side on which holes are
injected, becoming stronger as the sample is under-
doped (see Fig. 4). The underlying band structure is
not responsible since it is theoretically irrelevant and
experimentally implausible. In tunneling, a theorem
of Schrieffer removes much of the effects of quasi-
particle interactions, so that the broad spectra seen in
ARPES are referred back to the quasiparticle pole
energies; the «hump» structure in fact has a strong
resemblance to the incoherent part of the ARPES
EDC's. It is a remarkable achievement of «plain
vanilla» that it can give a sometimes quantitative
account of these spectra. In order to do so we must
modify the ansatz Eq. (5) for � in Eq. (4). BCS
functions are wave-packets in the space of total elect-
ron number and one makes up nonnumber-conserving
quasiparticles by taking advantage of this fact. This
grand canonical approach is justified because the
packet is centered at the correct particle number and
the amplitudes for N � 2, N and N � 2 are essentially
identical. But the projection process, while it does not
change particle number, does project out very differ-
ent numbers of states, so that after projection the
wave packet is skewed in N-space. In order to move
the center of the packet back to N, we must introduce
a fugacity factor dependent upon N:
� � �
� �g n n( )/2, (7)
and g turns out to be the familiar kinetic energy
renormalization factor 2 1x x/( )� . Although in (7) it
is clear that the factor g cannot change any energy
calculation since the Hamiltonian and projection con-
serve particle number, it is vital in understanding the
process of tunneling where a particle is added or
removed. Equation (7) may be rewritten by distribut-
ing the factors of g among the terms of the product,
appearing very different but actually this is an ob-
vious identity:
� � � � � � �� �� � � � � �
( )| (~ ~ )| ,† † † †u v c c u v c ck
k
k k k
k
k k k k
0 0
(8)
where ~u gu g u vk k k k� �/ 2 2 2 and ~v v g u vk k k k� �/ 2 2 2.
In (8), the ratio of probabilities of zero and single
occupancies is correct for the projected state and is
thus not altered by projection. What it makes clear is
that the projected state is made up from singlet pairs
in which the relative amplitude of paired holes (the u
term) is decreased relative to that of paired electron
spins (the v term) by the factor g. In a sense, there
are two types of condensed bosons, the valence bonds
of the RVB and the hole pairs, and in this theory we
set their relative amplitudes free, although they
remain coherent: they are «locked» together. The
principle on which we calculate the tunneling
spectrum is the following. Once we have chosen the
form (8) for �, we may define the single-particle
excitations whose energies satisfy the gap equations
in terms of the wave functions, Pci�� and Pci�
† �, or
equivalently, Pck�� and Pck�
† �, and these are now
equivalently normalized. But the matrix elements of
the tunneling process insert a particle or a hole prior
to the projection operation, at a particular site
effectively, so that they connect to the operators c Pi�
and c Pi�
† , and we have to commute the fermion
operator through the projection operator to determine
its effect.
We may write c Pc P c† † †( )� � �1 where ( )1 � P
projects onto states with a doubly-occupied site which
are effectively at infinite energy (after the canonical
transformation). Thus whenever the inserted particle
encounters an occupied site, the state is projected out,
and only with probability x does it encounter an
empty site, i.e., Pc†. But when it is Pc† it lands in a
legitimate excitation, i.e., Pc P Pc† †� . Thus when a
particle enters (with probability x) it does so co-
herently. The hole problem is less obvious. c may be
commuted through P with the result,
c P P n ci i i� � �� � ��( )1
� � �� �
�� �
�
�P
N
n c c ck
k
k
k k
k i( )†1
1
� � � �. (9)
The second term, when acting on �, is simply a
number times c: � � � ��n xi � ( )1 2/ . The third term is
386 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
P.W. Anderson
Bi Sr CaCu O2 2 2 3
–300 –200 –100 0 100 200 300
Bias, mV
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
d
I/
d
V
, a
rb
. u
n
its
Fig. 4. Tunneling spectrum in optimally doped BSCCO.
Data from S.H. Pan (unpublished).
genuinely incoherent, creating three excitations; but
these three can come from any energy in the spectrum
so we expect this term to be quite small everywhere
and to rise only as the square of the tunneling voltage
for small voltages. The net effect of (9), then, is that
cP
x
Pc�
�1
2
. (10)
Thus the ratio of the probabilities of tunneling of
electrons vs holes is (no surprise!) g x x� �2 1/( ). At
high energies � �� �, where the quasiparticles are
pure holes or electrons, this is the expected asym-
metry, and insofar as experiment is able to ascertain,
apparently this ratio agrees well (taking into account
the small error caused by the canonical transform-
ation exp ( )iS ).
The spectrum at lower energies is complicated by
the fact that superconducting quasiparticles are mix-
tures of electrons and holes. At the Fermi surface,
exactly at the gap energy, they are equal mixtures and
the singularity at the gap must be identical for the two
sides. The working out of the exact interpolation
formula for the tunnel current is a little complicated
and I give here only the formulas: the tunneling
density of states for electrons is
N E
d
dE
g
u
u v g
v
v u g
e ( , )� �
�
�
�
�
�
�
��
�
!
!!
� 2
2 2 2
2
2 2 2
. (11)
The g factor in this formula comes from the pro-
jection factor — which I emphasize does not multiply
the matrix element, it is essentially a relative number
of open channels. As we see, for v � 1, at high vol-
tage, the limiting value, 1, comes from the second
factor and the tunneling is suppressed by g. On the
other hand, for holes the tunneling density is
N E
d
dE
g
v
u v g
u
v u g
h ( , )� �
�
�
�
�
�
�
��
�
!
!!
� 2
2 2 2
2
2 2 2
. (12)
Here the g factor comes from the normalized fugacity
factor, and at high voltage u � 1, v � 0 and g cancels
out, giving the ratio g between the two limits. These
formulas fit data surprisingly well.
These are the formulas for fixed �. Note that for
� � 0, at the Fermi level, u v� and the two
are identical, the «coherence factor» amounting to
g g/ 1 2� . This agrees with sum rule arguments. The
asymmetry begins, however, with a vertical slope at
�, so cannot to be said to be exclusively a background
phenomenon, as is seen most clearly in the fact that
the peaks of observed spectra (see Fig. 4) appear to sit
on background levels of different heights. These for-
mulas must be integrated over the d-wave distribution
of gap values to give a prediction for comparison with
observed spectra. This we have done only roughly,
using P( )� �� �1 1 2/ , as though the Fermi surface
were circular and not taking into account the actual
band structure, which does somewhat affect the distri-
bution of � values. In Fig. 5, we give the predicted
spectra for a number of values of g, using this simpli-
fication.
The fit to experiment, at least in the main features,
is fundamentally significant. Of course, it helps con-
firm the basic structure of the theory, and the use of
superexchange as the major pairing interaction. But it
has even deeper implications. One is that even though
it is basically a mean-field theory based on an Har-
tree—Fock ansatz, it is not a Fermi liquid-based
theory, that is to say that in no way can it be
adiabatically continued to a BCS-like modification of
Fermi liquid theory. The most fundamental property
of Fermi liquid theory is hole-particle symmetry [22];
after all, how can one have a theory based on a
distribution of quasiparticles unless that distribution
counts particles minus holes, 1 for 1? This projective
theory has destroyed that symmetry in a very fun-
damental — yet simple — way.
Yet the projective feature is rooted in the real
physics of the system. As pointed out also by Capello
et al. [23], once one is above the Mott critical Uc,
there are what we called «anti-bound states» [24] and
what Ref. 22 calls «holon-doublon bound states»
which cannot be treated perturbatively but must
simply be projected out of the problem. A result is
Present status of the theory of the high-Tc cuprates
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 387
2.0
1.5
1.0
0.5
0
– 8 –6 –4 –2 0 2 4 6 8
Bias V/�max
d
I/
d
V
,
a
rb
. u
n
its
Z=0.3 z=0.2
0.1
0.05
0.03
Fig. 5. Predicted tunneling spectra for various Z g( )� .
that the spectrum is overcomplete and the particle
operators do not obey a simple fermion algebra. I
believe that the broad features seen in the mo-
mentum-resolved particle spectra are related to that
problem, but fortunately the tunneling spectrum is
simplified by the benign results of «Schrieffer’s
theorem» and is easier to interpret.
3.3. Fluctuations of the asymmetry parameter g
The asymmetry parameter, as I remarked, plays a
role similar to that of a condensate of hole pair bosons,
which is locked to the spinon pairs of the RVB by the
charge-spin locking process. Its average value is de-
termined by the doping and charge neutrality, but it is
evident that formally we can allow it to vary either in
space or in momentum space. More speculatively, we
can allow it a dynamic character, and I believe that its
collective modes do in fact give us extra degrees of
freedom which play a role in the non-Fermi liquid
behavior mentioned above. This and the other remarks
I will make in this rather speculative section are
inspired more by suggestions from experimental ob-
servations than by apriori theory, but I do believe that
treating g as a physical object can lead to considerable
insights.
The most obvious is the possibility of allowing g to
vary along the Fermi surface, just by making it a
function of k in (8). The mean value x g g� � � �/( )2
must be maintained for charge neutrality. The phe-
nomenon of «Fermi arcs» is observed by ARPES in
underdoped systems, where the regions of the Fermi
surface near the nodes remain sharply defined while
the antinodal regions smear out and disappear. The
nodal regions are also those where the kinetic energy
is greatest, so that one could gain energy by making g
large at the nodes and small at the zone corners. No
calculation of this proposed effect yet exists. A simp-
ler explanation for Fermi arcs, possibly equivalent,
has been found and will be published shortly.
It has also been proposed that the hole percentage
may vary spatially, in particular that at low doping g
could form a kind of charge density wave or «super-
conducting electron solid» [25]. The motivation could
be Madelung energy of the pairs; or it is possible that
there is a tendency to bistability near the low-doping
quantum critical point.
Finally, there is the question of phase fluctuations
of g (which is the appropriate variable to assign a
phase to, since it controls the charge carriers). It is
known that the phase transition at Tc is of «X–Y»
character both for optimal doping [15] and for very
low doping near the quantum critical point [26], and
that above Tc there is a large region in which the state
is best described as a «vortex liquid» rather than a
normal metal, i.e., there is a fluctuating supercon-
ducting order parameter (see Ref. 16). This has been
described as a regime in which � is still locked to the
kinetic energy but �, i.e., the phase of g, is freely
fluctuating [12]. There is a very important open
question here as to whether or not there is a transition
into a still higher T phase which has an RVB but is
not a vortex liquid [27].
4. Discussion: alternative approaches
The RMF theory works, based on t J� physics and
superexchange as the interaction, and can account
semiquantitatively for the basic phenomena of cuprate
superconductivity, and qualitatively for many more.
Why then are contradictory theories being promoted?
The most popular theories reject the Mott—Anderson
physics entirely and go in contrary directions. There
seems to be a psycho-social need among physicists for
an explanatory underline boson, some kind of tangible
glue to hold the pairs together, I suppose because of
the folk memory of talks about the BCS mechanism
and the analogy of two bodies on a mattress; or else a
simplified view of Feynman diagrams. It is felt, I
suppose, that the Mott theory is based on purely
repulsive forces — but those of us who actually
worked on BCS recognize that the phonon interaction
is not literally an attraction either, merely a partial
screening of the electrons’ Coulomb repulsion. Why a
superexchange integral universally agreed and expe-
rimentally measured to be of order 1000 degrees is
thought to be inadequate for pairing has always
escaped me; but it is. The two most popular glues are
phonons and antiferromagnetic spin fluctuations.
Phonons start out with a big disadvantage: the
BCS concept is irrevocably based on an on-site, local
interaction; and is incompatible with d-wave. In the
cuprates, the phonons are undoubtedly optical ones
involving the oxygen octahedron (oh, there are other
suggestions, but even less plausible) and there are
perhaps ways of distorting these in order to give a
d-wave, but I have never seen a plausible one. In-
trinsically, Einstein optical phonons lead to local
interactions. But, experiment is the best teacher. The
isotope effect measurements of Keller [28] find a
reasonably-sized isotope effect on Tc, apparently con-
firming the phonon hypothesis; but Keller was tho-
rough enough to also measure the isotope shift of � s ,
the superfluid density, " �# 2; and he finds that this
shifts by the same fractional amount. It was pointed
out very early in the game by Fisher et al. [29], that
unlike the polyelectronic metals for which BCS theory
works and the isotope shift comes entirely from the
pairing interaction, oxides are best understood as
tight-binding systems with interactions which depend
388 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
P.W. Anderson
exponentially on interatomic distances. Thus zero-
point vibrations will have an appreciable effect on
normal state properties such as the band mass which
determines � s . Since Tc is an X Y� transition as
already remarked, its value is expected to be directly
proportional to � s , the coupling in the xX Y� model,
as observed, so that apparently there is no experi-
mental isotope shift ascribable to the pairing interac-
tion. In fact, even if there were, Tc is insensitive to the
actual value of �, as we explained above. Extensive
ARPES studies have catalogued what may be phonon
effects on the quasiparticle dispersion [30] but these
seem to be irrelevant to the pairing mechanism. J, of
course, itself varies in a similar way as t with inter-
atomic distance and may provide a partial source of
the observed isotope shifts in dispersions. It seems
that calculating phonon effects, while worth doing for
its own sake, is not the most urgent task.
There are other phonon schemes, most notoriously
the bipolaron theory. One understands the impulse to
look this way, since polaron phenomena are so ubiqui-
tous in oxides. But very early on it became clear that
one reason the cuprates are so favored is that this case
is gloriously free of polaron effects, presumably beca-
use Cu� � and Cu� � � have similar Jahn—Teller dis-
placements. The remarkably detailed tunneling and
ARPES spectra demonstrating well-characterized qua-
siparticles exclude small polaron phenomenology. I
believe that Baskaran's theory [31] explaining the
electron-doped case as dominated by small polarons
must be essentially correct, and the contrast with hole
doping illustrates well what phenomenology polarons
might lead to.
A second putative source of the «glue» boson is
«antiferromagnetic spin fluctuations». This idea so-
unds similar to the Mott-based theory but is not at all
so, in fact proceeds on exactly the opposite principle:
that in the end the physics is to be obtained by
«summing all the diagrams» starting from a Fermi
liquid [32]. Another way to say it is that the assump-
tion is that the theory fits under the general scheme of
Ref. 10, where all interaction terms are renormalized
downwards, while the plain vanilla theory makes
the assumption that one must start by renormalizing
U �, with the Rice canonical transformation. I feel
that Uc, the Mott critical U, marks a fundamental
separatrix between basins of attraction, and that the
cuprate case is on the largeU side. The key question is
whether the frequency associated with most of the
pairing interaction is above a Mott—Hubbard gap,
and therefore cannot be represented by a boson whose
spectrum extends continuously to zero frequency. In
that case it might as well be represented by a simple
four-Fermion vertex J. The idea of antiferromagnetic
spin fluctuations is that the opposite is the case, and
that somehow if one can sum enough diagrams the
Mott gap will disappear from the problem and in-
teractions will proceed by the exchange of a putative
low-frequency spin-fluctuation boson.
Since, in fact, one cannot come close to summing
all the diagrams, papers based on this idea have
tended to contain about one parameter per expe-
rimental fact, and therefore to «explain» great num-
bers of these facts. Apparently recent advances in
experimental detail have led to exhaustion of in-
vention, and many rather crucial discoveries remain
unexplained, for example Fermi arcs, tunneling asym-
metry, the vortex liquid phase, the checkerboard,
Homes' identity.
There are a number of more mysterious suggested
sources for the «glue boson», many of which invoke
the equally opaque concept of a hidden «quantum
critical point»; their variety excludes detailed ex-
plication.
Perhaps no longer worthy of mention is the «stripe
theory», the problem of which was that it never
seemed to be a theory of the superconductivity, but
only a theory of the stripes themselves. Since stripes
are not common to many of the cuprate supercon-
ductors, and as time goes on to fewer and fewer, it is
hard to understand their relevance.
5. Conclusions and anticipations
Many of my conclusions were rather strongly sta-
ted in the Introduction. It seems that the Gutzwiller
method works perhaps even better than we had any
right to expect. It also has the added feature that it
brings out the deep difference in principle between a
Fermi-liquid based approach and the actual behavior
of the cuprates in a relatively simple and straight-
forward way, both in demonstrating the hole-particle
asymmetry of the Green's functions and in the «lock-
ing» phenomenon.
Quite understandably, there are other ways to
approach the same physical model, and some of them
have a good chance of being more accurate or rigorous
— for instance gauge and slave boson theories, one of
which I quoted here. One can certainly differ on the
applicability of the crude approximations made in
plain vanilla to make it soluble; and it is very
meaningful to try to add in further terms to the
interactions used, and to study the various accom-
panying phenomena such as coexistence with anti-
ferromagnetism. The major puzzle remains that of the
strange metal, the mysterious phase above T*, and the
strange quantum critical point where the d-wave gap
goes to zero. The linear T, linear in electron-electron
scattering mechanism which pervades the high-energy
Present status of the theory of the high-Tc cuprates
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 389
region is still a puzzle but must be a characteristic of
the purest Mott physics*.
I should not fail to mention the accumulation of
recent direct or semidirect calculational results all of
which are now tending to converge on the conclusion
that d-wave superconductivity undoubtedly appears in
the Hubbard and t J� models. I am sure these will be
represented well elsewhere in this volume; and of
course, I have absolutely no problem with them; it is a
matter of taste whether one prefers approximations
such as plain vanilla which allow understanding of the
phenomenology, or more exact but only semitrans-
parent calculations.
Throughout the paper I have alluded to avenues for
further exploitation of the method, specifically the
possible explanation of the «Fermi arcs» as a k-de-
pendence of g, and of nanoscale structures as spacial
modulations of it; but both will require more detailed
calculations than we are yet capable of.
6. Acknowledgements
I have first of all to acknowledge my long-term
collaborator Nai-Phuan Ong, for the innumerable
times he has helped me with experimental know-how
and theoretical comments. I should also acknowledge
my collaborators in the plain vanilla exercise, T.M.
Rice, P.A. Lee, Mohit Randeria, Nandini Trivedi and
Fu-Chun Zhang, as well as others who were involved
in finding the solution 17 years too early: Claudius
Gros and Gabi Kotliar. Others who have helped keep
my mind clear about experimental data have been
Doug Bonn, Nicole Bontemps, Bernhard Keimer, Sea-
mus Davis, Mike Norman, J.-C. Campuzano, Tom
Timusk, Kam Moler; this is only a tiny fraction of the
totality of individuals who have been helpful. But
none of the above need take any responsibility for
what I say here. Finally, there is my good friend and
sounding board, V. Muthukumar.
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390 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
P.W. Anderson
* The strange metal problem seems to have been solved, in very recent work (cond-mat/0512471). The phenomena turn out
to be surprisingly direct consequences of Gutzwiller projection.
|