A modern, but way too short history of the theory of superconductivity at a high temperature

0.23 MB·English

by  Jan Zaanen

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1 2.4A modern,but waytooshorthistoryofthetheoryofsuperconductivityatahigh
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4 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 PushingBCStothelimits: thespinfluctuationsuperglue. . . . . . . . . . . . . . 3

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1.3 ThelegacyofPhilipW.Anderson(I):Mottness. . . . . . . . . . . . . . . . . . . 8
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o 1.4 Thelegacyof PhilipW. Anderson(II):ResonatingValenceBondsanditsdescen-
c dants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
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r 1.5 Theorythatworks:thecompetingordersofthepseudogapregime. . . . . . . . . 16
p 1.6 Quantumcriticalmetalsandsuperconductivity. . . . . . . . . . . . . . . . . . . . 20
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2.4 A modern, but way too short history of the theory
of superconductivity at a high temperature.
JanZaanen
Instituut-LorentzforTheoreticalPhysics,LeidenUniversity,Leiden,TheNetherlands
1.1 Introduction ............................................................................ 1
1.2 PushingBCStothelimits:thespinfluctuationsuperglue. ................................ 3
1.3 ThelegacyofPhilipW.Anderson(I):Mottness. ......................................... 8
1.4 ThelegacyofPhilipW.Anderson(II):ResonatingValenceBondsanditsdescendants. .... 10
1.5 Theorythatworks: thecompetingordersofthepseudogapregime. ....................... 16
1.6 Quantumcriticalmetalsandsuperconductivity. .......................................... 20
1.1 Introduction
J.Zaanen-LeidenUniversity,TheNetherlands
Onecannotwritethehistoryofawarwhenitisstillragingandthisiscertainlyapplyingtothe
task we are facing dealing with the theory of superconductivity in the post BCS era. The BCS
theory was of course a monumentalachievement that deserves to be counted among the greatest
triumphsinphysicsofthetwentiethcentury.Withthediscoveryofhigh-T superconductivityinthe
c
cupratesin19861aconsensusemergedimmediatelythatsomethingelsewasatworkthantheclassic
(i.e. phonondriven)BCSmechanism. Aquarterofacenturylaterthissubjectisstillcontentious,
perhapsevenmoresothanatanyinstanceinthepast. Itisplainlyimpossibletowritean’objective’
history of any kind and this piece has no pretense in this regard. I thoughtit would be useful to
writeinsteadsomesortofeyewitnessaccount,servingtwopotentialreaderships: inthefirstplace
thenewcomerswhowonderabouttheprehistoryofsomestandardnotionsthatappearasestablished
andcontroversialatthesametime. Theothercustomersarerightnownomorethanapotentiality.
Imaginethat the theoryof high-T superconductivitywill turn truly gloriousat some pointin the
c
future. The era that I am discussing here will then have a similar status as the ’dark ages’ that
precededthediscoveryofBCStheoryasdescribedelsewhereinthisvolume. Icanimaginethatan
eyewitnessaccountlike the present one mightbe entertainingfor the professionalhistorianswho
wanttochroniclethistriumphof21-thcenturyphysics.
Iamawarethatthisexpositionisfarfromcomplete–therearequiteanumberofideasthatare
very interesting by themselves, while they might be on the long run more consequentialthan the
mattersIdiscusshere.However,giventhelengthrestrictionsIjusthavenotthespacetogiveatruly
1K.A.Mu¨ller,G.Bednorz,Z.Phys.B64(1986)189.
1
2 Superconductivity
comprehensiveoverviewofeverythingthathasbeenexplored. Ihavetorestrictmyselftoasketch
ofthehistorybehindthenotionsthatarerightnowmostontheforeground.Similarly,Idonotclaim
rigorwith regardto distributingthe creditsto the variousindividualsinvolved. Althoughatsome
instancesIwilldrawattentiontocontributionsthatIfindareunderexposed,thismaterialissurely
notintendedasbackgroundmaterialforawardcommittees.
Thiseyewitnessaccountisalsocoloredwithregardtothesubstancesthatarediscussed.Whether
we like it or not, humanityis a religiousspecies and belief systems play a big role when science
isstillinthemaking. OnebetterbeawareofitandinphysicsIliketobemyselfa’denomination
tourist’.Ihavedwelledaroundamongthevariousbeliefsystems,gettingintoittoadegreethatIfelt
myselftobeabeliever,tohopatthatpointtoanextsystem. InthecourseoftimeIhavesettledinto
a faith that is rightnow quite popular: the ’agnostic denominationin high-T superconductivity’.
c
Their dogma is as follows: ”the research in unconventional superconductivity is flourishing. It
is howeverowned by the experimentalists. Energizedby the high-T puzzle, the experimentation
c
in quantum matter leaped forward in the last twenty years, due to a variety of highly innovative
instrumentaldevelopments.Thesenewexperimentalfactshavehoweverjustdeepenedthemystery
forthe theorists. The agnosticzealis anchoredin the convictionthatthese fantastic new data are
beggingforan explanationin termsof mathematicalequationsof a beautyandelegancethatwill
measure up to the productsof Einstein and Dirac. Right now we have not the faintest idea what
theseequationsarebutthatshouldnotkeepusawayfromattemptingtofindthem.”
Surely this has imprinted on the organizationof this text. I will discuss matters in a sequence
thatisinahistoricalorder(fromoldtonew),whileatthesametimethesenseoftheawarenessof
dealingwithamystery(theagnosticdogma)isincreasing.
The pointof departureis the highlyinfluential’denomination’thatrests on the believethatthe
theoreticalframeworkasemergedinthe1950’sinessencestillsufficestoexplainthephysicsofthe
cupratesandsoforth.ThenormalstateisaFermi-liquidthatisperhapsabitobscuredbyperturba-
tiveprocesses. However,theessence ofBCS theoryisstill fullyintactandto explainthe high-T
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superconductivityoneneedsa ’superglue’thathasmorepunchthanthetraditionalphonons. This
superglueis than supposedto be embodiedby spin fluctuations. Since this ’fifties paradigm’has
beenaroundforalongtimeithasturnedintoaframeworkthatpretendstobequitequalitative.This
”holytrinity”(LocalDensityApproximation(LDA)bandstructure,RandomPhaseApproximation
(RPA)spinfluctuationsuperglue,Eliasbergequationsforthesuperconductivity)isrightnowinthe
middle of a revival in the context of the superconductivityin the iron pnictides. It surely should
betakenquiteseriouslybutmorethananywhereelsethedangerofreligiousillusionsisaroundthe
corner. InthisregardIperceiveitasbeneficialtobeawareofthelong,andcontentioushistoryof
thespinfluctuationsuperglueidea. Thisisthenarrativeofsection2.4.2.
Althoughithasbeenabitontheretreatrecently,thenextinfluentialschoolcanbecalled’RVB,
andrelatedphenomena’. Thisreferstoahighlyimaginativesetofideas,thatemergedinthewild
early years of high-T superconductivityunder the inspired intellectual leadership of Phil Ander-
c
son. Regardlesswhetherithasanythingtosayaboutthereallifeofhigh-T superconductorsthese
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deserveattentionifnotonlybecauseitdevelopedintoaquiteinterestingandinnovativebranchof
theoreticalphysics.
The spin fluctuation theoriesare rooted in the perturbativeschemes of the fermiologythat was
establishedinthe1950’s. TheResonatingValenceBond(RVB)ideaseventuallyrestontherecog-
nitionthatdopedMottinsulatorshavetobedifferent. Thisstorystartswiththelastskirmishesof
a war that is much older: the understanding of the Mott-insulating state itself. As I will discuss
in section 2.4.3, this pursuitstarted a long time ago, while it settled in the late 1980’swhen, as a
byproductofthehigh-T research,aratherdetailedandcompleteunderstandingofthereallifeMott
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insulatorswas established. Section2.4.4is dealingwith the RVB schoolof thought. Thisstarted
from an appealing idea by Phil Anderson that evolved during the wild early years of high-T in
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something of a mathematicalscience by itself. Differentfrom the 1950’s spin fluctuations, these
’gaugetheoriesofstronglycorrelatedfermions’restconceptuallyonthediscoveriesinthe1970’s
Amodern,butwaytooshorthistoryofthetheoryofsuperconductivityatahightemperature. 3
inhighenergyphysics.Althoughitstilllacksbothconvincingexperimentalsupportandrealmath-
ematicalcontrol,thesetheoriessurelyservewellthepurposeofconfusingthehighenergytheorists.
Atstakeisthemeaningofthegaugeprinciple. Infundamentalphysicsitisaprimordialprinciple.
The’RVB’gaugetheoriesareinsistinghoweverthatthegaugeprincipleisgovernedbycollectivity,
tellinghowlargenumbersof’un-gauged’microscopicdegreesoffreedomco-operateinunisonto
generate’gauged’collectivebehaviors.
Keepingthehistoricalorder,insection2.4.5Iwillfocusonthesubjectthathasbeendominating
the field duringthe last ten yearsor so: the physicsof the underdoped’pseudogapregime’. This
has been so much on the foregroundbecause it turned out that there was much to discover using
themodernexperimentaltechniques. Itisnowclearthatthereisacollectionofexoticcompeting
orders at work, including stripes, quantum liquid crystals and spontaneous diamagnetic currents.
Apparentlytheoreticalphysicsasweknowitdoesgiveagriponorderlythings,becausetheorists
didmakeabigdifferenceinthisparticularpursuit.
InthelastsectionIwilldiscusswhatmaybethebeginningofthehistoryofthefutureofhigh-T
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superconductivity.Thisrevolvesaroundtheideathatthequantumphysicsofthemysterioussuper-
conductorsisinthegripoftheoverarchingsymmetryprincipleofscaleinvariance.Inanalogywith
the classical criticalstate, this ’quantumcriticality’ is believedto be caused by a phase transition
happeningatzerotemperature,drivenbyquantumfluctuations. Giventheincreasedunderstanding
of the competing orders at work in the pseudogapphase this notion is at present becomingquite
popular in the cuprates. In the context of the heavy fermions it has been however obvious for a
long time that their superconductivityis closely related to quantumphase transitionsand at stake
is whether these share the essential physics with the cuprates. This is the stronghold for the ag-
nostictheoristsbecausenowhereelseitissoobviousthatwearelackingmathematicalmachinery
thatworksasinthecontextofquantumcritical’fermionic’metals. Iwillendthischronicleonan
optimistictonebyexpressingmyhopethatthestringtheoristsmightactuallyhavesuchmachinery
intheoffering.
1.2 Pushing BCS to thelimits: the spinfluctuation superglue.
The BCS theory is a remarkable achievement. A constant factor in the debate during the last
twentyfiveyearshasbeenwhetheritisatallpossibletogetaroundtheessenceofBCStoexplain
superconductivityat a high temperature. Its centralwheel is the Cooper instability which in turn
rests on the physicsof nearly free fermions. A first conditionis that the normalstate closely ap-
proachesthe Fermi-liquid fixed point near the superconductingtransition temperature. A second
conditionisthatthenetresidualquasiparticleinteractionsturnattractiveintheappropriatepairing
channel. AtthisinstancetheFermi-surfaceexertsitpower,bysinglingoutthepairingchannelas
uniquelysingular. Feedinga genericfermionsystem with a verystrongattractiveinteractionthat
overwhelmstheFermi-energywouldjustcauseaclumpingoftheparticlessincein(nearly)classical
systems thereis no specialstability associated with two particle boundstates2. The remainderof
BCS is simple. Cooperpairsare bosonsandthe CooperpairsBose condenseat themomentthey
form,ofcoursemodulothepossibilityof’dangerous’thermalorderparameterfluctuations.
ThequestioniswhetherthenormalstateissufficientlylikeaFermi-liquidfortheBCSmechanism
toapplydirectly. Thishasbeendividingthemindsmorethananythingelse–the’radical’theories
discussedinthenextsectionsdepartfromtheassertionthatthenormalstateissomethingelsethan
aFermiliquid. AssumingthegovernanceoftheFermi-liquid,theonlyissueisthentoexplainthe
2Althoughitisusuallyworkedundertherug,thisproblemhuntsideasinvolvinglocalpairing,likethebipolarons.
4 Superconductivity
originoftheattractiveinteractions. Thisisfarfromtrivialsinceelectronsactuallyrepeleachother
ratherstronglyonthemicroscopicscalethroughtheCoulombinteractions. Muchofthehardwork
ofdiminishingthesebareinteractionscanbeascribedtothemagicoftheFermi-liquid.Theubiqui-
tousnatureofthequasiparticlegasismuchmoreofamysterythanisoftenrealized.Theparticular
limitingcasewhereonthemicroscopicscaletheinteractionsaresubstantiallysmallerthanthebare
Fermienergyisnowadayswellunderstood,althoughthisinvolvesthefanciful’Polchinski-Shankar’
functionalrenormalizationgroup.Byprincipleonehastokeeptrackofaninfinityofcouplingcon-
stants’spanningup’theFermi-surface,whenintegratingoutshortdistancedegreesoffreedom. In
therenormalizationprocess,onefindsthatbesidesthe’harmless’marginalityoftheLandauFfunc-
tions, onehas onlyto watch a possible (marginal)relevancyin the Cooperchannel.3 The bottom
line is that with the exception of the BCS superconductivitythe Fermi-liquid is generically (i.e.,
awayfromspecialnestingconditions)stableasaninfraredfixedpoint.
Theproblemishoweverthatthesystemsofinterestinthepresenteraareinvariablyofthekind
wherethemicroscopicinteractionscaleexceedsthebarekineticenergybyorder(s)ofmagnitude.
One can no longer rely on the perturbative renormalization group and it is just an experimental
fact that Fermi-liquids are nevertheless ubiquitous. A classic example is 3He. At microscopic
distancesthisislikeaquantumversionofadense,highlycorrelatedVanderWaalsliquid.However,
uponcoolingit down to milli-kelvinsone findsa long wavelengthphysicscorrespondingwith an
impeccableFermi-gasof quasiparticles. The miracleis thatthe ’hardHelium balls’ havebecome
effectivelycompletelytransparentrelativetoeachother,onlycommunicatingviathePauliprinciple,
whiletheeffectsofinteractionsisthatthese’quasiHeliumatoms’arethreetimesheavierthanthe
realatoms. IlearnedfromRobertSchriefferthatin the1960’s-70’stheoriststriedhardtoexplain
thiswonder,notgettinganywhere.4 Fermi-liquidsareunreasonablyresilientandthisisperhapsthe
bestreasontotakethe’conservative’theoriesofhigh-T superconductivityserious.
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Thenextquestionbecomes:howtoturntheresidualweakquasiparticlerepulsionsoftheFermi-
liquid into serious attractive interactions? The classic BCS answer is of course phonons. The
exchange of bosons that are not governed by the gauge principle will generically give rise to an
inducedattractiveinteraction.Thephononsarejustperfect:theyareexternaltotheelectronsystem
since they originate in the lattice, their interactions with the quasiparticles emerge naturally and
theyhavetheaddedbenefitofcausingretardedinteractions. Phononenergiesaretinycomparedto
theFermienergyofconventionalmetalsandtheassociated’Migdalparameter’canbeexploitedto
simplifytheperturbationtheoryintoavirtuallyexact’resummed’Migdal-Eliashbergtheory.
Thiscontrolovertheclassicphonon-drivenBCStheoryexplainsthesociologicalearthquakethat
rumbledthroughthephysicscommunityfollowingtheBednorz-Mu¨llerdiscoveryofsuperconduc-
tivity at 34K in La2−xBaxCuO4 in 1986. Back then it was believed that phonons could not be
responsible for a T exceeding 30 K and a consensus emerged immediately that something new
c
was at work. In hindsight this is actually a bit embarrassing. It is a number game and numbers
aredangerous. Thestrongestclaimrestedonstabilityconsiderations5thatweresubsequentlychal-
lenged by theorists of Ginzburg’s high-T superconductivity group at the Lebedev institute6. In
c
fact, in 2001superconductivityat 40 K was discoveredin the simple MgB system. In no time a
2
consensusemergedthatthesuperconductivityisphonondriven,witha numberofmaterialdetails
(high phononfrequencies, multi-bands)conspiringto optimize the phononmechanism, while the
phononsresponsibleforthesuperconductivityareunrelatedtothefactorsrenderingthecrystaltobe
3ForafancifulrecentdiscussionseeS.A.Parameswaran,R.ShankarandS.L.Sondhi,arXiv:1008.2492.
4Remarkably,asaveryrecentdevelopmentitappearsthatthequantum-fieldsgravityholographicdualitiesofstringtheory
shedanewlightonthisoldproblem: theFermiliquidisrelatedtothegravitational stabilityofspecialblackholeswith
’fermionhair’,seesection2.4.5.
5M.L.CohenandP.W.Anderson,inSuperconductivityind-andf-bandMetals,ed.D.H.Douglass,AIP,NewYork(1972),
p.17.
6O.V.Dolgov,D.A.KirzhnitsandE.G.Maksimov,Rev.Mod.Phys.53(1981)81.
Amodern,butwaytooshorthistoryofthetheoryofsuperconductivityatahightemperature. 5
stable.Thisdevelopmentgavealsocredibilitytotheclaimthatsuchphonondrivensuperconductiv-
itycanbequantitativelystudiedusingLDAtypebandstructuretheory. ComputingT ’sfromfirst
c
principlesis very demandingand has only become possible recently by the increase of computer
power. MgB andanumberofothercaseshavelendcredibilitytothecasethatthesenumberscan
2
be trusted. This support the ’computational’case that 40K need not be the limit of conventional
phonondrivensuperconductivity7.
Despite these caveatsI am notaware ofanyexpertbelievingthatconventionalphononscan be
responsibleforthesuperconductivityincupratesorironpnictides. Perhapsthebestreasonisthat
the extreme electron-phonon coupling required to explain a T of 150K should somehow reveal
c
itself in band structure calculations, while these just indicate quite moderate couplings. This is
not to say that phononsdo not play any role. In fact, there are striking experimentalindications
that especially at very low dopingthe electron-phononcoupling is so strong that some variety of
small polaron physics is at work. This appears to invoke polar couplings that are ignored in the
LDAcalculationswhereimplicitlymetallicscreeningisassumed. Itcannotbeexcludedthateven
atoptimaldopingsuch a ’poormetal’polaronicphysicsis atworkandclaimsthatbipolaronsare
behindthesuperconductivitycannotbedismissedbeforehand.8
The main stream thinks however differently. Staying within the ”BCS paradigm” the quest to
explainthe high-T ’s turnsintothe issue whetheronecan identifynon-phononstuffthatcan take
c
theroleofthephonons,mediatingverystrongattractiveinteractions.Theonlyoptiononthetableis
tolookfor’glue’thatemergesfromtheinteractingelectronsystemitself. Therearequiteanumber
of options here, like the excitonic mechanism first proposed by Little, emphasizing special units
in the crystal structure characterized by a high electronic polarizability.9 Another old idea is the
Kohn-Luttingeroverscreeningeffect in 2D systems10 and the closely related idea of the plasmon
takingthe role of glue. However,by far the mostpopularview has beenup to the presentday to
pointatspinfluctuations.ThebasicideadatesbacktoBerkandSchriefferinthe1960’swhoaimed
atexplaningwhymetalsthatareclosetoatransitiontoanitinerantferromagneticstateusuallydo
notsuperconduct11. Theyintroduceda particularapproximatewaytocomputemattersthatforms
the conceptualcore of the myriadof improvedspin fluctuationtheoriesthathave been developed
sincethen.
A first hurdle is that in the electron system at microscopic distances there is no such thing as
a system of spins that has a separate existence of the electrons – the electrons just carry around
the spins themselves. The ’spin glue’ has thereforeitself to be a highlycooperativephenomenon
emergingat’large’distancesin the interactingelectronsystem. Uponapproachinga transitionto
amagneticstateinanitinerantmetalonedoeshoweverexpectprecursorsofthemagnetisminthe
paramagneticstate. Theseareintheformofshortlivedmagneticfluctuationsthatbecomeincreas-
ingly better defined upon approachingthe magnetic transition where they turn into real magnons
–theparamagnons. Usingtimedependentmeanfieldtheory(RPA)theparamagnonscanbecom-
puted perturbatively assuming weak interactions. Assuming in addition residual interactions be-
tweenthesespinfluctuationsandtheconductionelectrons,theirpropagatorscanbejustfeededinto
theMigdal-Eliasbergformalismtotaketheroleofphonons.
Theparticularbenefitofthisapproachisthatitisquitepredictivewithregardtothepairingsym-
metries.ForthecaseofferromagneticfluctuationsSchriefferandBerkshowedthatthesegenerically
causerepulsionsinthes-wavechannelexplainingwhynearlyferromagneticmetalsusuallydonot
7See, e.g., Marvin Cohen’s talk at the 2009 KITP superconductivity conference:
http://online.kitp.ucsb.edu/online/highertc09/cohen/
8A.S.AlexandrovandN.F.Mott,PolaronsandBipolarons,WorldScientific,Singapore(1995).
9Thisidearevivedinthecontextofpnictides:seeG.A.Sawatzkyetal.,Eur.Phys.Lett.86(2009)174409.
10W.KohnandJ.M.Luttinger,Phys.Rev.Lett.15(1965)524.
11N.F.BerkandJ.R.Schrieffer,Phys. Rev. Lett. 17(1966)433;animportantfollowupisS.DoniachandS.Engelsberg,
Phys.Rev.Lett.17(1966)750,focussingontherenormalizationoftheFermiliquid.
6 Superconductivity
superconduct. In the early 1970’s superfluidity in fermionic 3He was discovered. It was already
knownthatthenormalstateischaracterizedbyastronglyenhancedmagneticsusceptibilityindicat-
ingthattheHeliumFermi-liquidisonthevergeofbecomingferromagnetic.Giventhatthereareno
phononsaround, the idea that this superfluidityhad dealingswith paramagnonswas immediately
realized. Dealingwithphonons,theinducedattractiveinteractionsurelyfavorsaspinsingletorder
parameter,whileingoodmetalstheeffectivecouplingislocalizedinspaceandtherebymomentum
independent, favoring s-wave superconductivity. However, ferromagneticfluctuationsfavor spin-
tripletpairing. Thisinturnimpliesthatthepairsneedtobeinanorbitalstatewithunevenangular
momentum,havingthe addedbenefitthatthe real space pair wave functionhas to have a nodeat
theorigin. Thefermionsinthepairtherebyavoideachotherautomaticallysuchthatthehardcore
microscopic repulsions are circumvented. The bottom line is that although there might be a net
repulsionbetweenthequasiparticlesthismightwellturnouttosumupintoanattractiveinteraction
inahigherangularmomentumchannel.
The superfluidity in 3He is of the triplet variety and a discussion doing justice to the richness
of this subject deserves a separate chapter in a book like this.12 With regard to the microscopic
pairing mechanism a worthwhile lesson is the use of the Berk-Schrieffer logic by Anderson and
BrinkmantogetwithgreateffectthedetailsoftheA-phaseright.13 Thegrossmessageisthat3He
demonstratesthataninteractingfermionsystemasdominatedbymicroscopicrepulsiveinteractions
caneventuallyturnintoasuperconductor,withthepairingmechanismhavingundoubtedlydealings
with the spin fluctuations. However, the Anderson-Brinkmanresult is just showing that the spin
fluctuationscontributeto the attractive interactions, stabilizing a particular superconductingstate,
butitdoesnotprovethattheRPA style spinfluctuationsdoallthepairingworkexplaininginde-
tailwhyT iswhatitis. Forinstance,muchattentionwaspaidaswelltotheroleoftheattractive
c
tail of the microscopic Van der Waals interaction. Another example of this lack of microscopic
understandingis the questionwhetherthe mass enhancementsofthe 3He Fermiliquidcan be ac-
countedforbythecouplingtotheferromagneticspinfluctuationsorwhethertheyarerelatedtothe
proximity of the crystallization transition14. It appears that the quest to explain the nature of the
residualattractiveforceresponsibleforthepairingin3Heendedinastalemate. Manyfactorscan
contributeanditisjustanumbergame. Giventhattherearenocontrolledmathematicalmethods
thattellushowtoconnectthestronglyinteractingmicroscopytotheweaklyinteractinginfrared,it
isinfactplainlyimpossibletoaddressthissubtlequantitativematterwiththeavailabletheoretical
technologies. Withaneyeontheremainder,oneshouldbeawarethat3Heisinawaya bestcase
scenarioforglueideas,sincethenormalstateisawelldevelopedFermi-liquid,withasuperfluidity
thatisclearlyintheweakcouplinglimit,whilecomplicationslike(lattice)Umklappandphonons
areabsent.
Thenotionthatthesuperconductivityofelectronsystemsinsolidscouldbecausedbyintrinsic
electronicinteractionseffectsgotonthemainstageforthefirsttimebythediscoveryoftheheavy
fermionsuperconductorsinthelate1970’sand1980’s.Thegiganticmassrenormalizationsinthese
systemsleavenodoubtthatelectron-electroninteractioneffectsaredominant.Inadditionitbecame
gradually clear that the superconducting order parameters in these system are unconventionalby
default while magnetism is usually around the corner. During the last twenty years a consensus
emergedthatinthesesystemsthesuperconductivityiscloselyrelatedtothepresenceofaquantum
phasetransitionwhereatzerotemperatureamagneticallyorderedstategetsdestroyedbyquantum
fluctuations. Insection2.4.6Iwilltakeupthisthemeindetail,includingadiscussionoftheHertz
typetheoriesthatarecloselyrelatedtothespinfluctuations.
12Theclassic,hardtoimprovetreatiseisA.J.Leggett,Rev.Mod.Phys.47(1995)331.
13P.W.AndersonandW.F.Brinkman,Phys.Rev.Lett.30(1973)1108.
14D.Vollhardt,Rev. Mod. Phys. 56(1984)99;thisinturnrestedonthediscoverybyMauriceRiceandBillBrinkmanof
thedescriptionoftheMotttransitionintermsofGutzwillerprojectedwavefunctions:W.F.BrinkmanandT.M.Rice,Phys.
Rev.B2(1970)4302.TheseideaswereinturninspirationalinthedevelopmentoftheRVBtypetheoriesofsection2.4.4.
Amodern,butwaytooshorthistoryofthetheoryofsuperconductivityatahightemperature. 7
The spin fluctuation idea acquired directly prominency in the early history of cuprate high-T
c
superconductivity. Obviouslyan agentdifferentfromphononscausingthe superconductivitywas
atwork,whileinthefrenzyofthelate1980’sitbecameinnotimeclearthatcupratesuperconduc-
tivityemergedfromdopinganantiferromagnet. Inaddition,bothinelasticneutronscattering-and
NMR experimentsgave away that the metals at superconductingdoping concentrationsshow the
signaturesof pronouncedspin fluctuations. The spin-fluctuation-glueidea was lying on the shelf
anditwaspredictablytakenupbypartofthecommunity. ThemostvisibleadvocateswereDou-
glasScalapinoandDavidPines,representingsomewhatcomplementaryviewsthatarein2010still
discernible.Pinesembodiedthe’bottom-up’phenomenologicalapproach,analyzingingreatdetail
experimentalinformationonthespinfluctuations,aimingforaquantitativephenomenologicalde-
scription of the superconductivity. Scalapino approachedit from a more theoretical angle, trying
to cast the (incomplete) information from quantum Monte Carlo computations and so forth in a
diagrammaticframeworktayloredaccordingto the Berk-Schriefferframeworkand improvements
thereofflikethe”fluctuation-exchange”(FLEX)perturbativetheory.
Infact,thespinfluctuationadvocatesfacedaheavybattleintheearlyhistoryofhigh-T super-
c
conductivity. The predictivepower with regardto the pairing symmetrywas their stronghold. In
the presence of Umklapp associated with the scattering of the electrons againstthe crystal lattice
the pairingsymmetrystorybecomesmore interestingandspecific. The lattice warpsthe electron
dispersionsandtheFermi-surfaceshapewiththeramificationthatonthelevelofRPAcalculations
the spin fluctuations tend to get quite structured in momentum space. The cuprate fermi-surface
computedbyLDAbandstructure(andconfirmedbyphotoemissionmeasurements)shows’nesting’
featuresthatpromotesastrongenhancementofspinfluctuationsatlargemomenta.Thisinturnim-
pliesastrongmomentumdependenceofthepairinginteractionsmediatedbythesefluctuationsthat
stronglyfavorad superconductingorderparameter.Inthe1987-1990eraitwashowevertaken
x2−y2
forgrantedbythecommunityatlargethatcupratesuperconductivitywass-wave. Inhinsightthese
strongbelievesarenoteasy to explainonrationalgrounds: the datawere ofa poorqualityandit
appearstoberootedinsteadinaconservativesociologicalreflex–evend-wavewastoorisky.After
ahardfight,thatlastedforacoupleofyearsScalapinoandPinesetal. wereprovenrightwiththeir
d-waveclaim bythe phasesensitivemeasurementsofTsui, Kirtley,vanHarlingenandcoworkers
discussedelsewhereinthisbook. ThisisremarkableandIfinditrespectablethatthissuccesshas
motivateda stubbornattitude in the spinfluctuationcommunity: there hasto be somethingto the
notiongiventhatitgotthed-waveright. However,afteritbecameclearthatthepairingsymmetry
isd-waveinnotimeitturnedoutthatthisisalsothenaturalgroundstateinforinstancetheRVB
context (section 2.4.4), and even invoking the spontaneouscurrent phases of Varma as discussed
insection2.4.5. Althoughd-wavesuperconductivityisratherunnaturaldealingwithconventional
phonons,theconsensushasemergedthatitisjusttoogenericdealingwithelectronicmechanisms
forittoberegardedasconclusiveevidenceforanyspecificmechanism.
Atthismomentintime the ideaofa spin-fluctuation’superglue’isverymuchalive– counting
headsinthecommunityitmightbethemostinfluentialhypothesisalltogether.Partofthisappealis
surelyassociatedwiththefactthatitposesacrispchallengetotheexperimentalists: thesuperglue
islikeaholygrail,andonejusthastodigwiththemodernexperimentalmachineryofcondensed
matterphysicsintotheelectronmattertoisolateit. Factofthematterishoweverthataftertwenty
yearsof concertedeffortin thecupratesthe ’evidence’is asindirectas ithasbeenallalong. The
stateoftheartisperhapsbestrepresentedbyatourdeforcethatwaspublishedin2009.15 Empir-
icalinformationobtainedfromdirectmeasurementsof the spin fluctuationsbyneutronscattering
andtheelectronself energyeffectspickedupin photoemissioniscombinedwiththeresultsfrom
numericalcalculations,arrivingattheclaimthatthespinfluctuationgluecanfullyaccountforthe
superconductivity.Butthisworkalsorevealstheweaknessesofthis’paradigm’.Theevidencesare
15T.Dahmetal.,NaturePhysics5(2009)217.
8 Superconductivity
highlyquantitative,whileitdepartsfromoversimplifiedtoymodelswhereoneneverthelesslooses
mathematicalcontrolbecauseoftheintermediatestrengthofthecouplingsinvolved. Equallycon-
vincingcaseshaveappearedarguingthattheselfenergykinksinphotoemissionareentirelydueto
ratherweakelectron-phononcoupling.Themagneticresonancethatdominatesthespinfluctuation
spectrum in optimally doped cuprates can be interpreted rather as an effect than as the cause of
thesuperconductivity. Perhapsthemostdevastatingcounterevidenceavailableatpresentforsuch
aninterpretationisthestaggering’electroniccomplexity’oftheunderdopedpseudogapregimeas
discussedinsection2.4.5.Thesimplefermiology’RPAview’onthiscompetingorderappearsasa
grossoversimplificationofthisreality.
The adventof the pnictide superconductorsappeared initially as a shot in the arm for the spin
fluctuation idea. The LDA calculations indicate nesting properties of the Fermi-surface pockets
thatarebeneficialfortheRPAparamagnons,whiletheyareconsistentwiththe”(p ,0)”magnetism
foundintheparentcompounds.Thecasewasmadevigorouslythatsincetheparentsystemsarenot
Mottinsulatingithastobethattheironsuperconductorsarequiteitinerantandtypicalcandidates
forspinfluctuationsuperconductivity.Lastbutnotleast,apeculiar”s±”pairingsymmetrywaspre-
dictedbasedontheLDA-RPA-Eliashberg”trinity”.16 Itisnowwidelybelievedthatthissymmetry
isindeedrealizedinatleastsomeofthesesystems,althoughdefinitiveevidenceisstillmissing.
Muchoftheearlyexperimentalactivityinthisyoungfieldwasaimedattryingtofindevidencefor
thespin-fluctuationsuperglue.Remarkably,whenthemomentseemedfittocryvictorythebuilding
startedtocrumble. Dataappearedindicatingthatunexpected’nematic’phenomenaarehappening
in the underdopedpnictides that seem to parallel the pseudogapcomplexities of the cuprates. In
addition, the pnictide metals at optimal doping have weird properties like a linear resistivity that
againsuggeststhatthesehaveuncannycommonalitieswiththemysterystuffofthecuprates. This
is a very fresh development and it is much too early to say where it will land. However, I do
expectthatitisbeneficialforthenewcomersinthepnictidefieldtobeawareoftheverylongand
contentioushistoryofthespinfluctuationglueidea.
1.3 The legacyofPhilipW.Anderson (I):Mottness.
WheninsomefuturetherealhistoryofT canbewritten,theoddsarehighthatPhilipAnderson
c
willhavealegendaryroletoplay.TheinfluentialAndersoniantraditioninhigh-T superconductiv-
c
ityrevolvesaroundtheassertionthatthegroundrulesofmetalphysicsaslaiddowninthe1950’sdo
notapplytotheelectronsincuprates,forthereasonthathigh-T superconductivityemergesfrom
c
aMottinsulatorthatisdoped. Thisisinstarkcontrastwiththespin-fluctuationschoolofthought
asdiscussed in the previoussection. Thisdepartsfromthe notionthatonecan getaway with the
ideathatthenormalstateisinessenceaFermi-liquid. Althoughshakenhardbythepowerfulglue
bosons,thequasiparticlesformingaFermi-gasareeventuallyrulingthewavesanditisjustamatter
ofsortingouttheFeynmangraphsgoverningtheperturbationsofthequasiparticlegas.Whenhigh-
T was discoveredin 1986thiswastakenasa self-evidenttruismbynearlyeverybody. However,
c
Andersonpublishedoneofthefirst paperseveronthe subjectandthisis still themostinfluential
paperinthehistoryofthesubjectintroducingthe’resonatingvalencebond’(RVB)idea17 thatwill
be the focuspoint of the nextsection. This section is a preliminary: the success story of nailing
downthespecificsoftheparentcuprateMottinsulatorinthelate1980’s.
16Thisappearstohavebeenacloserace,withwinners: I.I.Mazinetal.,Phys. Rev. Lett. 101(2008)057003,beingcited
already500times.
17P.W.Anderson,Science235(1987)1196.