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High-T

Theory 451

and transport property is anomalous. To account for this, the marginal

Fermi liquid phenomenology was created.

It was proposed that a sin-

gle hypothesis about the particle-hole excitation spectrum can account for

all the observed anomalies. This spectrum determines the charge and spin

response, i.e. electronic polarizability and spin susceptibility, of the system

to external probes. The analysis of Ref. 29 was based on the experimen-

tal observation of an unusual Raman scattering spectrum that indicated a

polarizability P (q, ω), whose energy scale, rather than the Fermi energy as

in the usual case, was the temperature. Explicitly,

Im P (q, ω, T ) ∝ sgn(ω)



|ω|/T, |ω|  T

const, T  |ω|  ω

(5)

up to a cutoﬀ ω

It is very important to note that this form is characteristic of scaling

behavior of critical ﬂuctuations near a quantum critical point (“QCP”), to

which we shall return below. This was an early suggestion from theory that a

QCP, marked with a “Q” in Fig. 1, could control the normal state behavior

near optimal doping. Were superconductivity not to intervene below T

the QCP would govern a T = 0 transition between two phases. With the

assumption that P is only weakly dependent on wave vector q, the scattering

of electrons from bosonic ﬂuctuations of the above form leads to a linear-

in-T resistivity, which is perhaps the most striking observed property of the

strange metal phase near optimal doping.

The interaction between electrons and the bosons leads to an electron self

energy whose real part has a logarithmic singularity at zero temperature and

this produces a logarithmic singularity at the Fermi momentum in the single-

particle momentum distribution. This is to be contrasted with an ordinary

Fermi liquid that has a nonzero step discontinuity at the Fermi level and

with a generic “non-Fermi liquid” that has a power law behavior. This

logarithmic behavior is the origin of the term marginal Fermi liquid (MFL).

In the language of Green’s functions, the statement

is that quasiparticle

renormalization factor Z at the Fermi surface goes to zero as T → 0 as an

inverse logarithm:

G(k, ω) =

ω − 

− Σ(k, ω)

≈

ω − ˜

+ iΓ

, (6)

where the self energy is calculated from the interaction of the electrons

with the ﬂuctuations and is determined by a coupling constant λ and an

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452 E. Abrahams

ultraviolet cutoﬀ ω

MFL

(k, ω) ∼ λ



ω ln

− i



, x = max(|ω|, T ), (7)

so that

1/Z = (1 − ∂ ReΣ/∂ω) ∝ ln(ω

/T ), Γ ∝ max(ω, T ).

It is important to notice from Eq. (6) that the decay rate Γ from the in-

elastic scattering oﬀ the ﬂuctuations is linear in the electron excitation

energy, unlike the Fermi liquid where it is quadratic. In the MFL case,

then, the quasiparticles are only marginally deﬁned as the Fermi energy is

approached.

This form of electron propagator can be used to calculate various normal

state properties. Its imaginary part is the single-particle spectral function

that is measured in angle resolved photoemission (ARPES) experiments.

The marginal (i.e. logarithmic) behavior of the real part of the self energy

results in much more spectral weight appearing in the tail of the energy

distribution spectrum. The comparison to experiment is striking.

A cal-

culation

of the optical response gives an enhancement, compared to the

conventional Drude spectrum, of the spectral weight at high frequency, just

as seen in experiment.

There is an excellent review of the MFL phenomenology and its origins

in microscopic theory.

Elsewhere, there are several reviews

16,33

that give

further discussion of the applicability of the MFL phenomenology in both

the normal and superconducting states near optimal doping.

4.3. Stripes

The vast majority of theoretical attacks on the strong-correlation problem, of

which, as we have seen, the cuprate superconductors are a prime example,

have considered the physics of a homogeneous system. However, there is

no a priori reason why charge and/or spin inhomogeneities should not be

of fundamental importance. It is no secret that theoretical approaches to

the problems of the cuprates have often been the subject of controversy.

Perhaps none more so than the issue of the relevance of stripe inhomogeneity.

This subject has been recently reviewed

comprehensively, with complete

references to earlier overviews. Remarks here will be brief.

In the context of high-T

superconductors, stripe phases were ﬁrst

proposed

within a mean-ﬁeld treatment of the Hubbard model. Shortly

thereafter, numerical evidence

for electronic phase separation in the doped

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High-T

Theory 453

t−J model was found. That is, the doped holes were energetically favored to

segregate into a hole-rich region. It is clear why this may happen at low

doping: isolated holes destroy four magnetic bonds in a square lattice, while

clumped ones break fewer. Within the t−J model, it can be appreciated that

the doped holes might segregate in order to resolve the competition between

kinetic energy and superexchange. Beyond the t − J model, however, the

long-range Coulomb repulsion between holes disfavors such a charge accumu-

lation. The competition between the two eﬀects could lead to the formation

of an ordered array of stripes, in which the holes occupy charged lines in the

lattice. Between the lines, there are no missing spins, hence it is expected

that at low temperature, the spins will be ordered antiferromagnetically, as

in the undoped situation. From the theoretical perspective, stripe physics

is very attractive, as it involves essentially one-dimensional (1D) ribbons of

charge. Interacting electrons in 1D is a thoroughly studied problem and the

non-Fermi liquid behavior is well-known.

What about experimental evidence for stripes in the cuprates? Static

stripe order was ﬁrst observed

in 1995 by neutron scattering on a particu-

lar version of lanthanum-strontium cuprate that is doped with neodymium.

This discovery generated more theoretical activity on the picture described

above. The Nd substitution stabilizes a low-temperature tetragonal struc-

ture instead of the usual orthorhombic phase of LSCO. Neutron scattering

in the parent undoped compound shows commensurate magnetic peaks at

the antiferromagnetic ordering vector Q = (π/a, π/a).

In the Nd-doped

LSCO, four incommensurate magnetic peaks were observed displaced from

Q by shifts of magnitude δ. Furthermore, new charge Bragg peaks appeared

that were displaced from the usual lattice reﬂections by ±2δ. These fea-

tures are consistently accounted for by the formation of stripes along the

Cu-Cu directions in which holes aggregate along domain walls separated by

a distance π/δ. Between these, there is antiferromagnetic order of spins with

phase shift π across them. The major probes for observation of stripe phases

are neutron and x-ray scattering in the LSCO and YBCO compounds. In

addition, scanning tunneling microscope studies have revealed stripe and

checkerboard patterns in BSCCO. The latter technique, however, is strictly

a surface probe. Observation and interpretation are made diﬃcult by, among

other problems, the extent of quenched disorder, surface conditions, size of

single crystals.

Does stripe ordering in the cuprates have anything to do with supercon-

ductivity? While the physics of possible charge/spin stripe ordering may be

of fundamental interest, many claim that it is a distraction in the context

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454 E. Abrahams

of high-temperature superconductors. If one believes that the mechanism

for superconductivity in the electron-doped compounds,

such as NCCO,

is the same as that in the hole-doped materials, then the absence of any

experimental evidence for stripes in the former supports a view that they

are either irrelevant or detrimental. Furthermore, it is only in the LSCO

compounds where static stripe order has been seen and it is most prominent

at 1/8 doping, where it coincides with a pronounced dip in the supercon-

ducting dome of Fig. 1. That is, it is accompanied by a large suppression

of T

. Nevertheless, there are a number of proposals

that invoke stripe

ﬂuctuations as a source of pairing or as an enhancement of T

5. Pseudogap

The term pseudogap was invoked ﬁrst by Jacques Friedel,

who sought

to explain the lowering of T

on the underdoped side of the dome by the

development of a suppression of the density of states (pseudogap) as doping

was decreased. No aspect of the cuprate phase diagram has received more

theoretical attention in recent years than this pseudogap region, seen in

Fig. 1. Below a temperature T

∗

that can be appreciably higher than T

, a

suppression of the density of states is seen in most cuprates in the under-

doped region. This is observed in a variety of experiments, in nuclear mag-

netic resonance, Raman scattering, single-electron spectroscopy, transport,

and in thermodynamics.

As shown in the ﬁgure, T

∗

decreases as doping is

increased, opposite to the behavior of T

, and it ultimately disappears into

the superconducting dome near or above optimal doping.

Whether the T

∗

line represents a true thermodynamic phase transition,

as is proposed in some theories of competing orders, or rather a crossover is

still an open question. Distinct thermodynamic features have not (yet?) been

found, but there are classes of phase transitions that do not exhibit the usual

speciﬁc heat singularity. It has also been argued that the presence of disorder

may round a transition suﬃciently to make an underlying singularity diﬃcult

to observe.

ARPES experiments have been particularly informative about the physics

of the cuprates since they reveal the momentum dependence of the Fermi

surface, the superconducting gap and the pseudogap in the Brillouin zone.

Signiﬁcantly, the symmetry of the pseudogap in momentum space is d-wave-

like, just as the superconducting gap. However, there is an important dif-

ference. Whereas the superconducting gap is zero only along lines in the

(±π, ±π) (“nodal”) directions in the Brillouin zone, the pseudogap vanishes

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High-T

Theory 455





/

(b)

,

,



(a)

Fig. 4. (a) Typical 2D Brillouin zone for hole-doped cuprates. Black line: Normal

state Fermi line. Red dot: Nodal point where superconducting gap is zero. Aqua

line: Fermi arc where pseudogap vanishes. (π, π) = nodal direction. (π, 0) = antin-

odal direction. (b) Sketch of angle dependence of gaps. Red: Superconducting gap

(cos 2θ). Aqua: Pseudogap and Fermi arc. Gaps are normalized to their respective

maximum values.

along an arc of the Fermi surface, Fig. 4(a). The length of the arc is temper-

ature dependent and extrapolates to zero at T = 0.

Thus, in the antinodal

region a gap in the spectrum exists, but along the arcs there are coher-

ent quasiparticle excitations. A complete analysis of the pseudogap physics

must account as well for the origin of this “Fermi arc.” The momentum

dependence of the gaps is depicted in Fig. 4(b).

This striking phenomenon is unprecedented; it appears nowhere else in

condensed matter physics and has been a source of enormous theoretical

activity, ranging from speculation to microscopic calculation. In fact, almost

all theoretical activity at the time of writing of this chapter appears to be

focused on one or another aspect of the pseudogap phenomenon. Many

scenarios for the pseudogap have been proposed. Most fall into one of four

classes: preformed pairs, RVB, competing orders, onset of incoherence.

As is the case with most of the unusual experimental observations on these

fascinating materials, several theoretical scenarios often seem to account for

results in the pseudogap region. For example, it is observed that the in-plane

conductivity is little aﬀected by the formation of the pseudogap while the

c-axis conductivity normal to the planes is signiﬁcantly suppressed. Both

the preformed pairs and the RVB pictures give simple explanations of this

observation, see below.

5.1. Preformed pairs

It is a commonplace that ﬂuctuations of an order parameter can have

important eﬀects on properties above a phase transition. In the cuprates,

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456 E. Abrahams

such ﬂuctuations are likely to be important because of low dimensional-

ity (quasi-two-dimensions) and the short coherence length in the supercon-

ducting state. Consequently, it was quite natural to propose that Cooper

pairs formed at T

∗

but without the long-range phase coherence required for

superconductivity, and that these would be in a singlet state with d-wave

symmetry, thus a precursor to superconductivity. Of course, one could well

argue, as some have,

that the pseudogap is instead a precursor for the

Mott insulator at zero doping.

ARPES experiments have become very important for understanding the

pseudogap, since they discriminate between the nodal and antinodal regions

of the Fermi surface, whereas the other probes generally give the overall den-

sity of states. Perhaps the most important evidence for the preformed pairs

scenario comes from relatively recent ARPES experiments

that reveal, in

the pseudogap phase, a Bogoliubov quasiparticle spectrum, symmetric about

the Fermi level, in the antinodal region and a normal quasiparticle spectrum

in the region of the Fermi arc, where of course there is no pseudogap. In this

description, the nodal region has an arc rather than a nodal point as in the

superconducting state because in a nonzero region around the nodal point

the pseudogap is small and is destroyed by quasiparticle damping.

In this picture, the suppression of the c-axis charge transport at T

∗

simply a consequence of band structure: the interplane hopping vanishes at

the nodal points where the quasiparticles carry the in-plane current, while it

is large in the antinodal region where the pseudogap opens, thus suppressing

the interplane transport.

5.2. RVB and the pseudogap

We have already seen the RVB phase diagram in Fig. 3 that the RVB theory

can give an account of crossovers, as a function of doping, from gapped

state (pseudogap) to strange metal to Fermi liquid. Here, the pseudogap is

essentially the energy required to break the spin singlets that constitute the

RVB spin-liquid state below the blue dashed line in the ﬁgure.

On the issue of the c-axis conductivity versus the in-plane transport

referred to above, the RVB picture is simple: the singlet pairing of the neu-

tral spinons does not aﬀect the in-plane charge transport. For the c-axis

conductivity the transport between planes must be by the full electrons

(or rather, holes) so that spinon and holon need to be recombined, which

requires breaking spin singlets and paying the spin gap energy. In fact,

all the single-carrier probes, such as photoemission, require the cost of the

spinon spin gap energy.

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High-T

Theory 457

This appealing picture derives from a mean-ﬁeld treatment of the RVB

state. However, low dimensionality and strong coupling suggest that ﬂuctu-

ation eﬀects are important. Consequently, there have been many extensions

and elaborations of the RVB theory, often in the language of gauge theory.

These are comprehensively reviewed in Ref. 17, along with a discussion of

many experimental results.

An RVB alternative to the gauge theory approach is based on the orig-

inal Anderson idea

of the projected wave function, see Sec. 3. Analytic

treatment of the projection is diﬃcult so that much of what has been pub-

lished is computational. Apart from computational approaches, a mean-

ﬁeld theory

based on the Gutzwiller projection and using the Gutzwiller

approximation

of assigning weight factors to terms in the Hamiltonian was

introduced very early and it is reviewed in Ref. 21. This has more recently

been used to formulate an ansatz for the self energy

in the pseudogap

regime. This gives an opportunity to make analytic analyses of pseudogap

physics. A review of this theory with an application to ARPES experiments

and pertinent references is given in Ref. 48.

5.3. Competing orders

A quite diﬀerent scenario for the pseudogap phenomenon is that it is a

consequence of a transition to an ordered phase that competes with super-

conductivity. The T

∗

line is then a thermodynamic phase transition line

that would terminate in a T = 0 quantum critical point, were it not for the

development of superconductivity at T

. As in Friedel’s original conjecture,

since the competition gets stronger the less the doping, this accounts for the

suppression of the superconducting T

as the doping is reduced into the un-

derdoped regime. The competing orders that have been discussed include

spin and charge density waves, including stripes and two types of circulating

current phases. A brief description of the latter group follows.

5.3.1. d-density wave

The d-density wave (DDW) state is one in which there is staggered orbital

current order across the plaquettes of a square lattice, see Fig. 5(a). First

discussed in 1989

and 1991

and revived and renamed in 2000–2001,

and elaborated in a number of papers since by Chakravarty and cowork-

ers.

In Ref. 51, the authors proposed a d-density wave phenomenology to

characterize the pseudogap region of the cuprate phase diagram. As can be

seen from the ﬁgure, the state breaks parity, time-reversal symmetry and the

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458 E. Abrahams

(b)

(a)

Fig. 5. Circulating current patters in (a) d-density wave order and (b) loop-current

order. Plus (+) and minus (−) denote magnetic ﬂux out or into the paper.

lattice translational symmetry, thereby doubling the unit cell. Therefore, it

is to be expected that there will be a reconstruction of the original hole-like

Fermi surface [Fig. 4(a)] and the new Fermi surface can have both electron

and hole pockets.

The commensurate DDW order parameter is a particle-hole spin singlet,

as it is in the familiar charge density wave (CDW) case, but with a d-wave

form factor instead of s-wave.

∝ hc

k+Q,α

k,α

i = iW

= iW

(cos k

− cos k

), Q = (π, π).

Where does the DDW state come from? It can arise from a mean-ﬁeld

treatment of some underlying Hamiltonian, which in mean ﬁeld has the form

H =

k,α

{(

− µ)c

k,α

+ [iW

(cos k

− cos k

k+Q,α

k,α

+ h.c.]}

Here, 

is the electron band structure and 2W

is the amplitude of the result-

ing DDW gap in the quasiparticle spectrum, which now has two branches:





+ 

k+Q

(

− 

k+Q

)

+ 4W



(8)

As with most of the theoretical ideas about the cuprates, the DDW pseu-

dogap scenario is controversial. It can account, perhaps not uniquely, for

many observations in the underdoped region. This is summarized in Ref.

52. While the DDW phenomenology is concerned with the pseudogap and

not with the mechanism of superconductivity, it should be appreciated that

the competition of DDW and d-wave superconductivity is very direct and

explicit: They compete for the same portions of the Fermi surface. This

implies that as doping decreases and T

∗

goes up, the superconducting T

goes down, as observed.

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High-T

Theory 459

Perhaps the most compelling evidence for a Fermi surface reconstruc-

tion in the pseudogap region comes from recent quantum oscillation exper-

iments

that measure Fermi surface areas. While there are still conﬂicting

reports on the details, the simplest interpretation is that multiple Fermi

pockets are observed. This implies breaking of the lattice translational sym-

metry. This would be consistent with a (possibly incommensurate) DDW

phase. However, direct observation of DDW order via spin-polarized neutron

scattering has so far returned mixed results.

5.3.2. Loop currents in the CuO

lattice

A diﬀerent form of a circulating current phase was ﬁrst proposed in 1997

by C.M. Varma, who has published a thorough review of this theory of

the pseudogap phase.

The loop-current state is obtained in a mean-ﬁeld

treatment of an extended Hubbard model of the form of Eq. (1). Several

requirements motivated the choice of eﬀective Hamiltonian and the nature of

the mean-ﬁeld solution. The fact that charge ﬂuctuations on the oxygens of

the CuO

planes are not negligible leads one to consider a three-band [as in

Fig. 2(b)] extended Hubbard model

[Eq. (1)] built from the p

, p

orbitals

on oxygen and the d

−y

orbital on copper. The success of the marginal

Fermi liquid phenomenology (discussed in Sec. 4.2) makes it likely that the

underlying ﬂuctuations of Eq. (5) are associated with a quantum critical

point underneath the superconducting dome in the region of optimal doping,

“Q” in Fig. 1. The phase on the underdoped side of the QCP is responsible

for the pseudogap. To extend the picture, one requires that the critical

ﬂuctuations P of Eq. (5), which exist in the quantum critical region near

optimal doping, are those of the order parameter of the new phase; and to

complete the picture, it is supposed that exchange of these same ﬂuctuations

provide the pairing interaction that produces d-wave superconductivity.

An important issue is whether the pseudogap phase breaks the lattice

translation symmetry, as it does in the DDW scenario, above. Recent ob-

servations that bear on this issue are the magneto-oscillation experiments,

which can be interpreted by a Fermi surface reconstruction arising from

a breaking of the translation symmetry. An alternate explanation

that

in fact preserves the translation symmetry is based on a form of magnetic

breakdown. If there is no broken lattice translation or rotation symmetry

in this pseudogap phase, there remains the possibility that the new phase is

characterized by broken time reversal and parity. For suﬃciently large ionic

Cu-O interaction, V in Eq. (1), a mean-ﬁeld solution for this Hamiltonian

gives stable states with circulating loop currents within a unit cell, such that

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460 E. Abrahams

lattice translation symmetry is preserved. The ﬁrst experimental indication

of the existence of such states came from ARPES.

The loop-current state

of Fig. 5(b) was shown

to be consistent with the experimental data.

Since the ﬁrst observation reported in Ref. 57, other experiments

diﬀerent classes of underdoped cuprate superconductors, using ARPES, po-

larized neutron scattering and magnetization measurements, have given evi-

dence for a time-reversal breaking but translation preserving order below the

pseudogap temperature T

∗

. The local orbital moments however appear not

be perpendicular to the CuO

planes, which means that a current pattern

may be more complicated, perhaps involving the oxygens out of the plane.

The loop-current picture is then the following: guided by the success of

the MFL phenomenology, its indication of the existence of a quantum criti-

cal point and various experimental constraints, a mean-ﬁeld solution of the

extended three-band Hubbard model gives a stable loop current phase. In

order to demonstrate the gap in the electron excitation spectrum, it is nec-

essary to derive the coupling of the electrons with the (bosonic) ﬂuctuations

of the loop currents. This is a very complicated procedure and is carried out

in Ref. 54. The result is an anisotropic gap in the excitation spectrum at

T = 0 of an unusual form:





+ D(k), E

> µ



− D(k), E

< µ



, D(k) ∼ cos

(2θ). (9)

Here, θ is the angle on the Fermi surface as deﬁned in Fig. 4. This pseudogap

is tied to the Fermi level and has similarities to a d-wave gap (∼ cos 2θ) but

lacks the latter’s sign (phase) changes around the Fermi surface. At nonzero

T , the point nodes of this gap broaden into arcs.

5.4. Antinodal incoherence

A major diﬃculty of the strong-correlation many-body problem discussed

in Sec. 1 is a shortage of comprehensive analytic calculational tools to ana-

lyze the relevant eﬀective Hamiltonians. The dynamical mean ﬁeld theory

(DMFT)

and its cluster and cellular extensions

have been developed to

treat situations in which dynamical correlations are dominant. In the con-

text of the cuprate superconductors, DMFT has been applied primarily to

the one-band Hubbard model and to the t − J model. As a matter of fact,

cluster DMFT methods applied to the one-band Hubbard model including

next-neighbor hopping have given the essential features (antiferromagnetism,

superconductivity, pseudogap) of the cuprate phase diagram as a function

of temperature and doping.

62,63