Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

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21 Concepts in High Temperature Superconductivity

E.W. Carlson Purdue University, Dept. of Physics, IN, USA

V. J . E m e r y Brookhaven National Laboratory Dept. of Physics, New York, USA

S.A. Kivelson Stanford University, Dept. of Physics, Ca, USA

D. Orgad The Hebrew Univ. of Jerusalem, Racah Institute of Physics, Israel

Preface ................................................................................1227

21.1 Introduction ...........................................................................1227

We highlight our main themes: mesoscale structure and the need for a kinetic energy driven mechanism.

21.2 High Temperature Superconductivity is Hard to Attain .....................................1230

We explore the reasons why high temperature superconductivity is so difficult to achieve from the

perspective of the BCS-Eliashberg approach. Because of retardation, increasing the f requency of the

intermediate boson canno t significantly raise T

. Strong coupling tends to reduce the phase order ing

temperature, and promote competing instabilities.

21.2.1EffectsoftheCoulombRepulsionandRetardationonPairing..........................1230

21.2.2PairingVersusPhaseOrdering......................................................1231

21.2.3CompetingOrders................................................................1232

21.3 Superconductivity in the Cuprates: General Considerations .................................1234

Some of the mostimportantexperimental facts c oncerning the cuprate high temperature superconductors

ar e described with particular emphasis on those which indica te the need for a new approach to the

mechanism of high temperature superconductivity. A perspective on the pseudogap phenomena and the

origin of d-wave-like pairing is presented.

21.3.1AFermiSurfaceInstabilityRequiresaFermiSurface..................................1234

21.3.2ThereIsNoRoomforRetardation ..................................................1235

21.3.3PairingisCollective!...............................................................1235

21.3.4WhatDeterminestheSymmetryofthePairWavefunction?............................1236

21.3.5WhatDoesthePseudogapMean?...................................................1238

21.4 Preview: Our View of the Phase Diagram ..................................................1244

We brieﬂy sketch our view of how the interplay between stripe and s uperconducting order leads to high

temperature superconductivity, various pseudogap phenomena, and non-Fermi liquid behaviors that

resemble the physics o f the 1D electron gas. This serves as a trailer for Sect. 21.13.

21.5 Quasi-1D Superconductors ..............................................................1245

The well developed theory of quasi-one dimensional superconductors is introduced as the best theoretical

laboratory for the study of strongly correla ted electron ﬂuids. The normal state is a non-Fermi liquid,

in which the electron is fractionalized. It can exhibit a broad p seudogap regime for temperatures above

but below the high tempera ture Tomonaga-Luttinger liquid regime. T

marks a point of dimensional

crossover, where familiar electron quasiparticles appear with the onset of long range superconducting

phase coherence.

21.5.1ElementaryExcitationsofthe1DEG.................................................1246

21.5.2SpectralFunctionsofthe1DEG–SignaturesofFractionalization........................1250

21.5.3DimensionalCrossoverinaQuasi-1DSuperconductor................................1253

21.5.4AlternativeRoutestoDimensionalCrossover.........................................1256

1226 E.W. Carlson et al.

21.6 Quasi-1D Physics in a Dynamical Stripe Array .............................................1257

An interesting generalization of the quasi-1D system occurs when the background geometry on which

the constituent 1DEG’s reside is, itself, dyna mically ﬂuctuating. This situation arises in conducting stripe

phases.

21.6.1OrderinginthePresenceofQuasi-staticStripeFluctuations...........................1257

21.6.2TheGeneralSmecticFixedPoint....................................................1259

21.7 Electron Fractionalization in D > 1 as a Mechanism of High Temperature Superconductivity . . 1259

Spin-charge separa tion seems to o ffer an attractive route to high temperature superconductivity. It occurs

robustly in 1D, but is now known to occur in higher dimensions as well, although seemingly only under

very special circ umstances.

21.7.1RVBandSpin-ChargeSeparationinTwoDimensions.................................1260

21.7.2 Is an Insulating Spin Liquid Ground State Possible in D > 1?...........................1261

21.7.3TopologicalOrderandElectronFractionalization.....................................1262

21.8 Superconductors with Small Superﬂuid Density ...........................................1262

In contrast tothe case of conventional superconductors, in superconductors with small superﬂuid density,

ﬂuctuations of the phase of the superconducting order parameter affect the properties of the system in

profound ways.

21.8.1 What Ground State Properties Predict T

?............................................1262

21.8.2AnIllustrativeExample:GranularSuperconductors...................................1263

21.8.3ClassicalPhaseFluctuations........................................................1267

21.8.4QuantumConsiderations..........................................................1269

21.8.5ApplicabilitytotheCuprates.......................................................1271

21.9 Lessons from Weak Coupling ............................................................1272

The weak coupling renormalization group approach to the Fermi liquid and the 1DEG is presented. The

role of retardation, the physics of the Coulomb pseudopotential, and the nonrenormalization of the elec-

tron phonon coupling in a BCS superconductor are systematically derived. The strong renormalization

of the electron-phonon interaction in the 1DEG is contrasted with this—it is suggested that this may be

a more general feature of non-Fermi liquids.

21.9.1 Perturbative RG Approach in D > 1.................................................1272

21.9.2 Perturbative RG Approach in D =1.................................................1275

21.10 Lessons from Strong Coupling ...........................................................1277

In certain special cases, well controlled analytic results can be obtained in the limit in which the bare

electron-electron and/or electron-phonon interactions are strong. We discuss several such cases, and in

particular we demonstrat e a theoretically well established mechanism in one dimension, the “spin gap

proximity effect,” by which strong repulsive interac tions between electrons can result in a large and robus t

spin gap and strongly enhanced local superconductingcorrelations. We propose this as the paradigmatic

mechanism of high temperature superconductivity.

21.10.1TheHolsteinModelofInteractingElectronsandPhonons............................1277

21.10.2InsulatingQuantumAntiferromagnets.............................................1279

21.10.3TheIsolatedSquare..............................................................1282

21.10.4TheSpinGapProximityEffectMechanism..........................................1284

21.10.5InhomogeneityforSuperconductivity..............................................1286

21 Concepts in High Temperature Superconductivity 1227

21.11 Lessons from Numerical Studies of Hubbard and Related Models ............................1286

The careful use of numerical studies to understand the physics on scales relevant to the mechanism of

high tempera ture superconductivity is advocated.

21.11.1PropertiesofDopedLadders......................................................1287

21.11.2 Properties of the Two-Dimensional t − J Model.....................................1295

21.12 Doped Antiferromagnets ................................................................1300

There are many indications that the cuprate superconductors should be viewed as doped antiferromag-

netic insulators. The motion of dilute holes in an ani terromagnet is highly frustrated, and attempts to

understand the implications of this problem correspondingly frustrating. However, one generic solution

is macroscopic or microscopic phase separation into hole poor antiferromagnetic and hole rich metallic

regions (where the hole motion is unfrustrated).

21.12.1FrustrationoftheMotionofDiluteHolesinanAntiferromagnet......................1300

21.12.2CoulombFrustratedPhaseSeparationandStripes...................................1305

21.12.3AvoidedCriticalPhenomena......................................................1307

21.12.4TheCupratesasDopedAntiferromagnets...........................................1308

21.12.5AdditionalConsiderationsandAlternativePerspectives..............................1309

21.13 Stripes and High Temperature Superconductivity ..........................................1310

We present a coherent view—our v iew—of high temperature superconductivity in the cuprate s upercon-

ductors. This section is more broadly phenomenological than is the rest of this paper.

21.13.1WhereDoStripesOccurinthePhaseDiagram? .....................................1311

21.13.2StripeCrystals,Fluids,andElectronicLiquidCrystals................................1315

21.13.3OurViewofthePhaseDiagram—Reprise...........................................1317

21.13.4SomeOpenQuestions............................................................1320

List of Symbols .........................................................................1326

References.............................................................................1327

Preface

In this paper, we discuss the core theoretical issues

associated with the mechanism of high temperature

superconductivity. Our primary focus is on physics

at intermediate temperature scales of order T

(as

well as the somewhat larger “pseudogap” tempera-

ture) and energies of order the gap maximum, 

in the context of rigorous theoretical analysis. We

also brieﬂy discuss these ideas in the experimental

context of the cuprates, and close with a set of open

issues. Note, high temperature superconductivity is

difficult to achieve, even in principle. It requires that

the energy scales of both pairing and phase coher-

ence be simultaneously elevated, yet model calcula-

tions and empirics reveal an inherent competition

between the two. Furthermore, in a highly correlated

system, the electronic structure is prone to local in-

homogeneity (and therefore also to competing or-

ders). However, such mesoscale structure can miti-

gate the deleterious effects of the Coulomb repulsion,

while providing a novel route to superconductivity

through kinetic-energy driven pairing.

21.1 Introduction

The virtues of BCS theory are extolled.

Conventional superconductors are good metals in

their normal states, and are well described by Fermi

1228 E.W. Carlson et al.

liquid theory. They also exhibit a hierarchy of en-

ergy scales, E

 !

 k

,whereE

and !

are the Fermi and Debye energies, respectively, and

is the superconducting transition temperature.

Fermi energy !

Debye frequency k

Boltzmann’s

constant Moreover, one typically does not have to

think about the interplay between superconductiv-

ity and any other sort of collective ordering, since

in most cases the only weak coupling instability of a

Fermi liquid is to superconductivity. These reasons

underlie the success of the BCS-Eliashberg–Migdal

theory in describing metallic superconductors [1].

The assumptions of BCS theory are violated by the

high temperature superconductors.

By contrast, the cuprate high temperature supercon-

ductors [2] (and various other newly discovered ma-

terialswithhigh superconductingtransitiontemper-

atures) are highly correlated“bad metals,”[3,4] with

normal state properties that are not at all those of

a Fermi liquid. There is compelling evidence that

they are better thought of as doped Mott insula-

tors, rather than as strongly interacting versions of

conventional metals [5–7]. The cuprates also exhibit

numerous types of low temperature order which in-

teract strongly with the superconductivity,the most

prominent being antiferromagnetism and the unidi-

rectional charge and spin density wave“stripe”order.

These orders can compete or coexist with supercon-

ductivity. Furthermore, whereas phase ﬂuctuations

of the superconducting order parameter are negligi-

bly small in conventional superconductors, ﬂuctua-

tion effects are of order one in the high temperature

superconductors because of their much smaller su-

perﬂuid stiffness.

Apparently, none of this complicates the funda-

mental character of the superconducting order pa-

rameter: it is still a charge 2e scalar field, although it

transforms according to a nontrivial representation

of the pointgroupsymmetry of thecrystal—itis a“d-

wave superconductor.”At asymptotically low temper-

atures and energies, there is every reason to expect

that the physics isdominated by nodal quasiparticles

that are similar to those that one might find in a BCS

superconductor of the same symmetry. Indeed,there

is considerable direct experimentalevidence that this

expectation is realized [8–11].However,the failure of

Fermi liquid theory to describe the normal state and

the presence of competing orders necessitates an en-

tirely different approach to understanding much of

the physics, especially at intermediate scales of order

, which is the relevant scale for the mechanism

of high temperature superconductivity.

The pur pose of this paper.

It is the purpose of this paper to address the physics

of high temperature superconductivity at these in-

termediate scales. We pay particular attention to

the problem of charge dynamics in doped Mott in-

sulators. We also stress the physics of quasi-one-

dimensional superconductors,in part because that is

the one theoretically well understood limit in which

superconductivity emerges from a non-Fermi liquid

normal state. To the extent that the physics evolves

adiabatically from the quasi-one to the quasi-two-

dimensionallimit,this caseprovidesconsiderablein-

sight into the actual problem of interest. The sound-

ness of this approach can be argued from the obser-

vation that YBa

7−ı

(YBCO) (which is strongly

orthorhombic) exhibits very similar physics to that

of the more tetragonal cuprates. Since the conductiv-

ity and the superﬂuid density in YBCO exhibit a fac-

tor of 2 or greateranisotropy within the plane [12,13],

this material is already part way toward the quasi-

one-dimensional limit without substantial changes

in the physics! In the second place, because of the

delicate interplay between stripe and superconduct-

ing orders observed in the cuprates, it is reason-

able to speculate that the electronic structure may

be literally quasi-one-dimensional at the local level,

even when little of this anisotropy is apparent at the

macroscopic scale.

Mesoscale electronic structure is emphasized.

A prominent theme of this article is the role of

mesoscale structure [14]. Because the kinetic energy

is strongly dominant in good metals,their wavefunc-

tions are very rigid and hence the electron density is

highly homogeneous in real space, even in the pres-

ence of a spatially varying external potential (e.g.

disorder). In a highly correlated system, the elec-

tronic structure is much more prone to inhomo-

21 Concepts in High Temperature Superconductivity 1229

geneity [15–17], and intermediate scale structures

(stripes are an example) are likely an integral piece

of the physics. Indeed, based on the systematics of

local superconducting correlations in exact solutions

of various limiting models and in numerical “exper-

iments”on t − J and Hubbardmodels, we have come

to the conclusion that mesoscale structure may be

essential to a mechanism of high temperature super-

conductingpairing.(See Sects.21.10 and 21.11.) This

is a potentially important guiding principle in the

search for new high temperature superconductors.

A kinetic energy driven mechanism is called for.

This is related to a concept that we believe is cen-

tral to the mechanism of high temperature super-

conductivity: the condensation is driven by a lower-

ing of kinetic energy. A Fermi liquid normal state is

essentially the ground state of the electron kinetic

energy, so any superconducting state which emerges

from it must have higher kinetic energy. The energy

gain which powers the superconducting transition

from a Fermi liquid must therefore be energy of

interaction—this underlies any BCS-like approach

to the problem. In the opposite limit of strong re-

pulsive interactions between electrons, the normal

statehashighkineticenergy.Itisthuspossibleto

conceive of a kinetic energy driven mechanism of su-

perconductivity, in which the strong frustration of

the kinetic energy is partially relieved upon enter-

ing the superconducting state [18–24]. Such a mech-

anism does not require subtle induced attractions,

but derives directly from the strong repulsion be-

tween electrons. As will be discussed in Sect. 21.10,

the proximity effect in the conventional theory of su-

perconductivity is a prototypical example of such a

kinetic energy driven mechanism: when a supercon-

ductor and a normal metal are placed in contact with

each other, the electrons in the metal pair (even if the

interactions between them are repulsive) in order to

lower their zero point kinetic energy by delocalizing

across the interface. A related phenomenon, which

we have called the“spin gap proximity effect”[20,25]

(see Sect. 21.10.4), produces strong superconduct-

ing correlations in t − J and Hubbard ladders [26],

where the reduction of kinetic energy transverse to

the ladder direction drives pairing. It is unclear to

us whether experiments can unambiguously distin-

guish between a potential energy and a kinetic en-

ergy driven mechanism.

But since the interaction

between electrons is strongly repulsive for the sys-

tems in question, we feel that the a priori case for a

kinetic energy driven mechanism is very strong.

The plan of the article is discussed.

Our approach in this article is first to analyze var-

ious aspects of high temperature superconductivity

as abstract problems in theoretical physics, and then

to discuss their specific application to the cuprate

high temperature superconductors.

We have al so

attempted to make each section self contained. Al-

though many readers no doubt will be drawn to read

this compelling article in its entirety, we have also

tried to make it useful for those readers who are inter-

ested in learning about one or another more specific

issue. The first eleven sections focus on theoretical

issues, except for Sect. 21.4, where we brieﬂy sketch

the mechanism in light of our view of the phase di-

agram of the cuprate superconductors. In the final

section,we focusmore directlyon the physics of high

temperature superconductivity in the cuprates, and

summarize some of the experimental issues that re-

main, in our opinion,unsettled.

 =1

=1.

Except where dimensional arguments are impor-

tant, we will henceforth work with units in which

 = k

=1.

Recent papers by Molegraaf et al. [27] and Santadner-Syro et al. [28] present very plausible experimental evidence of

a kinetic energy driven mechanism of superconductivity in at least certain high temperature superconductors.

While examples of similar behavior can be found in other materials, for ease of exposition we have focused on this

single example.

1230 E.W. Carlson et al.

21.2 High Temperature Superconductivity

is Hard to Attain

Catch 22

Superconductivity in metals is the result of two dis-

tinct quantum phenomena: pairing and long range

phase coherence. In conventional homogeneous su-

perconductors, the phase stiffness is so great that

these two phenomena occur simultaneously. On the

other hand,in granular superconductors and Joseph-

son junction arrays,pairing occurs at the bulk tran-

sition temperature of the constituent metal, while

long range phase coherence, if it occurs at all, ob-

tains at a much lower temperature characteristic

of the Josephson coupling between superconduct-

ing grains. To achieve high temperature supercon-

ductivity requires that both scales be elevated simul-

taneously. However, given that the bare interactions

between electrons are strongly repulsive, it is some-

what miraculous that electron pairing occurs at all.

Strong interactions, which might enable pairing at

high scales, typically also have the effect of strongly

suppressing the phase stiffness, and moreover typi-

cally induce other orders

in the system which com-

pete with superconductivity.

BCS is not for high T

superconductivity.

It is important in any discussion of the theory of

high temperature superconductivity to have clearly

in mind why conventional metallic superconductors,

which are so completely understoodin the context of

the Fermi liquid based BCS-Eliashberg theory, rarely

haveT

’s above15K,andneverabove30 K.Inthissec-

tion, we brieﬂy discuss the principal reasons why a

straightforwardextension of the BCS-Eliashberg the-

ory does not provide a framework for understanding

high temperature superconductivity, whether in the

cuprate superconductors, or in C

,orpossibly even

BaKBiO.

21.2.1 Effects of the Coulomb Repulsion and

Retardation on Pairing

In conventional BCS superconductors, the instanta-

neous interactions between electrons are typically

repulsive (or at best very weakly attractive)—it is

only because the phonon induced attraction is re-

tarded that it (barely) dominates at low frequencies.

Even if new types of intermediate bosons are invoked

to replace phonons in a straightforwardvariantof the

BCS mechanism, the instantaneous interactions will

still be repulsive, so any induced attraction is typi-

cally weak, and only operative at low frequencies.

Never forget the Coulomb interaction.

Strangely enough, the deleterious effects of the

Coulomb interactionon high temperature supercon-

ductivity has been largely ignored in the theoret-

ical literature. The suggestion has been made that

high pairing scales can be achieved by replacing

the relatively low frequency phonons which medi-

ate the pairing in conventional metals by higher fre-

quency bosonic modes, such as the spin waves in

thehigh temperaturesuperconductors[29–32] or the

shape modes [33,34] of C

molecules. However, in

most theoretical treatmentsof this idea,the Coulomb

pseudopotential is either neglected or treated in a

cavalier manner. That is, models are considered in

which the instantaneous interactions between elec-

trons are strongly attractive. This is almost certainly

[14,20,35–37] an unphysical assumption!

In Sect. 21.9, we use modern renormalization

group (RG) methods [38,39] to derive the conven-

tional expression for the Coulomb pseudopotential,

and how it enters the effective pairing interaction

at frequencies lower than the Debye frequency, !

This theory is well controlled so long as !

 E

and the interaction strengths are not too large. It is

worth reﬂecting on a well known, but remarkably

profound result that emerges from this analysis: As

electronic states are integrated out between the mi-

croscopic scale E

and the intermediate scale,!

,the

electron-phonon interaction is unrenormalized(and

so can be well estimated from microscopic consider-

ations), but the Coulomb repulsion is reduced from

a bare value, , to a renormalized value,



∗

= /[1 +  log(E

)]. (21.1)

Here,asistraditional, and 

∗

are the dimension-

less measures of the interaction strength obtained

That is, magnetic, structural, etc.