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The cluster extensions of DMFT beyond the original single-site formu-
lation permit an analysis that distinguishes different regions of momentum
space, in particular, for understanding of the pseudogap, the nodal and
the antinodal regions of the Brillouin zone. A cluster DMFT treatment
64
of the Hubbard model at large but finite on-site interaction U implies a
scenario for the formation of the pseudogap and the Fermi arc in the un-
derdoped regime. In the antinodal region, the strong short-range correla-
tions generated by the effective exchange coupling (as in RVB and the t −J
model) result in the suppression of quasiparticle coherence in the antinodal
region of the zone and the appearance of a pseudogap. At the same time,
in the nodal regions, quasiparticle coherence is preserved and a Fermi arc
remains over a nonzero region of the original Fermi surface, whose antinodal
parts are destroyed by the pseudogap. All this is caused by the short-range
part of the Coulomb interaction, since for smaller values of U, the entire
Fermi surface is metallic. An extension
65
of this work to the behavior of the
c-axis (interplane) transport in the underdoped regime gives support to this
interpretation and gives good agreement with experimental results referred
to above, just before Sec. 5.1.
5.5. Some open questions on the pseudogap
At the time of this writing, there are several aspects of the pseudogap phe-
nomenon that have had conflicting experimental reports. Once sorted out,
these features will impose important constraints on theories of the origin of
the pseudogap and of the mechanism for superconductivity in the cuprates.
“One gap or two.” This controversial issue refers to the nature of the gap
in the superconducting state. Based on the observation that the pseudo-
gap energy scale T
∗
and the superconducting energy scale T
c
behave in an
opposite fashion with doping concentration, it has been suggested that the
gap in the antinodal region is the pseudogap persisting below T
c
and that
the nodal region has the true superconducting gap i.e. the latter appears
on the Fermi arc below T
c
. The one-gap scenario describes a single d-wave
gap that persists outside the Fermi arc above T
c
, as described briefly in
Sec. 5.1. The preformed pairs scenario is a “one-gap” picture and most of
the other theories give two gaps. Experiments give conflicting information
on this issue.
Theories differ on whether the pseudogap is pinned to the Fermi level as
it would be for preformed pairs. For example, as we have seen, the DDW
gap can be centered above the Fermi level, while the loop currents gap is on
the Fermi level. ARPES experiments have given conflicting results.
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462 E. Abrahams
We have also seen that the shape of the pseudogap can be different for
different theories. Hopefully, future high-resolution experiments will shed
light on this question.
6. Quantum Criticality
The quantum critical phenomenon has become a major topic in condensed
matter research in recent years, especially in the context of the heavy-fermion
compounds.
66
What about the cuprates? An early theoretical suggestion
of the existence of a putative quantum critical point (“QCP”) under the
superconducting dome (“Q” in Fig. 1) came from the marginal Fermi liquid
theory.
29
On the experimental side, Loram, Tallon and coworkers
67
proposed
the existence of a T = 0 critical point near or slightly above optimal doping.
As with almost everything else about the cuprates, the QCP issue is
controversial. There is conflicting evidence about the position of the T
∗
line
with respect to the superconducting dome. Does it enter the dome near
optimal doping and terminate at a QCP as shown in Fig. 1 or does it merge
with the overdoped side of the dome and introduce no new QCP. Further
disagreement arises on the question of whether, if it exists at all, the QCP has
anything to do with the superconductivity. Scenarios of competing orders
involve a QCP that separates the pseudogap phase from a Fermi liquid,
while for theories based on RVB ideas, i.e. proximity to the Mott insulator,
quantum criticality is only a distraction: there is no QCP in the RVB phase
diagram, Fig. 3.
As mentioned earlier, a QCP is an essential ingredient of the loop-current
theory that leads to marginal Fermi liquid. The loop-current transition can
be mapped
54
onto an quantum Ashkin-Teller model, whose correlator of
quantum fluctuations has the form, Eq. (5), of the fluctuation susceptibil-
ity originally hypothesized for the marginal Fermi liquid phenomenology.
29
This gives the MFL picture a microscopic basis and makes clear the role
of the quantum critical point. The loop-current theory then is based both
on the existence of a QCP near optimal doping and on the three-band ex-
tended Hubbard model. However, it is not certain that this route to MFL is
unique. A calculation
68
using the dynamic cluster approximation extension
of dynamical mean field theory to treat the one-band Hubbard model with
moderate U/t = 6 and 8 gives evidence of a QCP. A MFL region is found
that appears to emanate from a QCP at about 15% doping. The region
is bounded on the low doping side by a T
∗
line determining the onset of a
pseudogap and by a crossover to a Fermi liquid on the high doping side, just
as in Fig. 1.
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In addition to the possible QCP near optimal doping, there are, of course,
other QCPs as shown in the phase diagram, Fig. 1. A system that is tuned
away from a Mott insulator by doping will exhibit a QCP that determines the
metal-insulator transition and the pseudogap region could be interpreted
17
as the quantum critical region associated with this QCP. Such a QCP, inter-
preted as an “antiferromagnetic quantum critical point” (AFQP) has been
discussed in the context of the spin-fluctuation phenomenology in Ref. 28b
and also within a phenomenological two-fluid description.
69
In this latter
work, an examination of various experimentally determined magnetic quan-
tities leads the authors to propose a scaling behavior reflecting the AFQP
that involves two components — a spin liquid of localized spins whose exci-
tations couple to a Fermi liquid. This generates a gap that is identified with
the experimentally observed pseudogap.
An in-depth discussion of cuprate quantum criticality can be found in
Ref. 34, Sec. VI-B. As they are also an essential ingredient in the theory of
fluctuating stripes, quantum critical fluctuations are reviewed in Ref. 37.
7. Superconductivity
The reader will have noticed that little of this review concerns the super-
conducting state of the cuprates. This is because a generalization of BCS
theory to the d-wave symmetry of the order parameter has done very well
to elucidate most of the physical properties of the superconducting state in
the optimally doped and overdoped regions of the phase diagram, without
of course revealing the mechanism of pairing. Indeed, as is reflected in this
review, the most striking mysteries of the cuprates are found in the unusual
normal state properties.
Theoretical ideas about the mechanism for superconductivity in the
cuprates fall into two groups. One group deals with the pairing in the super-
conducting state as analogous to that of conventional superconductors; thus
pairing mediated by some bosonic excitation, the “glue.” The other group
is based on the idea that the superconductivity arises out of the insulating
Mott state upon doping and is not to be described by the pairing of coherent
quasiparticles, although coherence may be restored below T
c
. Much of the
discussion here involves the question of whether the one-band Hubbard and
t − J models with realistic parameters contain the principal features of the
cuprate phase diagram, in particular d-wave superconductivity.
There is now ample evidence, mostly from cluster dynamical mean field
approaches, that the one-band Hubbard and t − J models do describe
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464 E. Abrahams
the important properties of the cuprate superconductors, including d-wave
superconductivity, the pseudogap at low doping below a T
∗
that decreases
with doping, the phenomenon of Fermi arcs, the competition between super-
conductivity and antiferromagnetism, evidence for a quantum critical point
beneath the superconducting dome.
DMFT methods rely heavily on numerical computation, but they have
advantages over earlier standard computational techniques for strongly
correlated systems, such as quantum Monte Carlo and exact diagonaliza-
tion on small clusters. The DMFT methods, embedding as they do a finite
size cluster in a self-consistent bath, are relevant for the thermodynamic
limit. Furthermore, since the basic construction of DMFT is analytic and
treats self energies explicitly, it gives an avenue to describing the underlying
physics of the results obtained. For superconductivity in the cuprates, these
results are described in Refs. 62 and 63 and can be summarized as follows.
At large values of the Hubbard on-site repulsion U ≥ 12t,
62
the under-
doped regime is characterized by a pseudogap in the one-particle spectrum
and can be understood as holes moving in a sea of RVB singlets. To the
right of the T
∗
line (Fig. 1), the optimally doped regime, there are no co-
herent quasiparticles as a result of a very large scattering rate, i.e. Im Σ.
Coherence is restored upon entering the superconducting state, at least
in the nodal region. There is a crossover to a strongly-correlated Fermi
liquid in the overdoped region. The main contribution to the superconduct-
ing condensation energy is the effective superexchange, that is, the equivalent
of the J term in the t −J model. The superexchange as the pairing mecha-
nism was already anticipated in early RVB theory, where for example in Ref.
24, both the d-wave superconductivity and the pseudogap were predicted.
At moderate values of U,
63
of order 8t, studies using the dynamical cluster
approximation extension of DMFT have also reproduced the main features
of the cuprate phase diagram. This method has also been used to determine
the irreducible particle-particle scattering vertex Γ in the Hubbard model.
Interestingly, the frequency dependence of the d-wave part of Γ reflects the
frequency dependence of the spin susceptibility at the antiferromagnetic wave
vector (π, π). This lends some support to phenomenologies based on spin-
fluctuation mediated pairing.
28
The bosons that have been invoked to be the pairing glue include phonons,
stripe fluctuations, spin fluctuations, and loop-current fluctuations. Here,
comments will be made on the last two of these. In both cases, unlike
the phonon-mediated case, the bosonic variables are composites made up
from the same degrees of freedom as the electrons that are being paired.
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Then the calculations that are based on the familiar Eliashberg equations
70
used for conventional superconductors are actually semi-phenomenological in
nature as they involve coherent quasiparticles interacting with the relevant
independent bosonic degrees of freedom.
71
The case of spin fluctuation
exchange pairing has already been discussed in Sec. 4.1 and mentioned in
the previous paragraph.
As mentioned earlier in Secs. 5.3.2 and 6, the fluctuations of the loop-
current order are given by a susceptibility of the form of P in Eq. (5). These
fluctuations are then invoked and calculated as the bosonic pairing glue for
superconductivity. It is necessary to determine the structure of the coupling
of electrons to these fluctuations as input to the Eliashberg procedure. This
has been carried out recently
72
and d-wave superconductivity is obtained.
Using parameters obtained from the fit of MFL to normal state transport,
the authors find reasonable values for T
c
and the gap ratio ∆/T
c
.
On the face of it, the strong-coupling and Eliashberg approaches stand in
sharp contrast. In the former, the physics is determined by the competition
between the superexchange, set by J and the kinetic energy of the doped
holes, set by xt, where x is the hole concentration. The latter is essentially
a weak-coupling formulation that phenomenologically treats the bosons and
electrons as separate entities.
8. Conclusion
Why is the cuprate problem hard? The easy answer is that because of
the simultaneous occurrence of strong coupling, low dimensionality and low
spin, all conventional approaches such as perturbation theory and mean-field
theory are inadequate. Perhaps it is a consequence of this confluence of dif-
ficulties that the cuprates are distinguished by having a number of different
competing states close in energy, depending upon the level of doping. Fur-
thermore the many unusual properties, e.g. pseudogap, strange metal, Fermi
arcs, that have been discovered experimentally are quite unprecedented; this
puts severe constraints on candidate theories for high-temperature supercon-
ductivity. Nevertheless, arguments can and have been made that the high-T
c
problem is solved (“it depends on what you mean by ‘solved’”).
There seem to be two different fundamental answers to the question of
what is it about the cuprates that is responsible for their high transition
temperatures. One is that they are quasi-two-dimensional with spin 1/2 and
they are proximate to a Mott insulator with its accompanying antiferromag-
netic exchange interactions, which happen to be unusually strong. Then
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466 E. Abrahams
the conclusion is that the basic physics can be understood by studying the
one-band two-dimensional Hubbard and t − J models, which contain these
elements. This approach of course originates in Anderson’s seminal 1987
paper
2
and based on the ideas contained there, d-wave superconductivity
and pseudogap were already predicted in 1988 in a slave boson mean field
calculation.
24
It has been stated
73
that that already represents the solution.
Since that early work, the same underlying physics has motivated further
developments. These include gauge theories and cluster extensions of
DMFT, which, as described in earlier sections, have elaborated many of the
features of the cuprates. A more complete solution?
In sharp contrast is the view that what is important is the chemistry
of the cuprates and the uniqueness of the energy scales associated with the
two-dimensional CuO
2
lattice. Thus, a three-band model is considered es-
sential,
54,69
in which the on-site Hubbard U may simply be treated in mean
field. This is the approach taken, for example, in the loop-current theory,
54
which in its turn also describes many of the features of the cuprates. Another
complete solution?
Setting aside the perhaps disagreeable situation of having too many solu-
tions rather than none, one should recognize the important contributions of
cuprate superconductivity to both experimental and theoretical condensed
matter physics. Remarkable advances have been made in materials prepa-
ration and in experimental techniques. On the theory side, while a BCS
d-wave theory may be serviceable in the superconducting state, its origins
in the normal state remain controversial. In fact, the normal state, in its
various manifestations as a function of doping (pseudogap, strange metal,
Fermi liquid crossover), gives every indication of being the consequence of
a new state of matter. This strong-correlation problem continues to gen-
erate entirely new theoretical frameworks, concepts and predictions that of
necessity go beyond the Landau paradigms of coherent quasiparticles and
ordinary phase transitions. In the years to come, this new physics will be
applied to newly discovered correlated materials and will continue to domi-
nate the frontier of condensed matter science.
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