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PhysicsofDopedMott-HubbardInsulatorsandtheHigh-TemperatureSuperconductorsClaudiusGrosJ.-W.GoetheUniversityFrankfurt1transitiontemperatures 1910193019501970199020102030020406080100120140277KYearRoomtemperatureOHgLiquidLiquidnitrogenOPbONbCONb3SnONb3GaOYBaCuOOBiCaSrCuOOLaSrCuOOLaBaCuOOHgCaBaCuOONbONbNOTlCaBaCuO??hydrogen1986:Discoverybyofhigh-temperaturesuperconductivitybyBednorz&M¨uller2high-temperaturesuperconductorsnoveltysuperconductivityinadopedoxide(badmetal)dopedMott-HubbardinsulatordoAFandSCcompeteorcooperate?3BCSsuperconductivityBardeen-Cooper-Schrieffer⊲standardtheoryfor‘conventional’superconductors.⊲superconductivityˆ=paircondensation|ΨBCSi=∏kuk+vkc†k↑c†−k↓|0iu2k/v2k=121±ξkqξ2k+Δ2k⊲one-particledispersionξk=¯h2k22m−μBogoliubovquasiparticles•dispersionEk=qξ2k+Δ2k•gapΔk≡Δ(s-wave)Δ=0Δ0[Edegger,Muthukumar,Gros‘06]4d-wavesuperconductivitygap-functionΔk=Δ(coskx−cosky)⊲nodesalong(1,1)direction++++----00.511.5200.511.52N(ω)ω/Δ[Fang.et.al.(2005)]•predictedbytheory⊲correlation-inducedsuperconductivity•experimentalverification⊲phase-sensitiveinterference⊲tunneling,ARPES5keyissue:thepseudogapwhathappensforTTc?dopingxcAFTd−waveSCT*(x)T(x)pseudogapΤcscΔΔΔ(Τ)T*04590SCPGspectralgapFermisurfaceangle0101NSPGSCkykx•one-particleexcitations⊲transport,ARPES⊲pseudoΔk≈Δ(coskx−cosky)•Fermiarc6BCS-ratio•universalforweak-coupling2ΔkBTc=(3.52s-wave4.3d-wave[Kugler,Fischer,Renner,Ono,Ando‘01]ΔΔSTM05101520253035012345678T*/TCkTBC2kT*=4.3Bp2p//•high-temperaturesuperconductors2ΔkBT∗=4.37dopedMott-HubbardinsulatorsH=−t∑i,j,σc†iσcjσ+c†jσciσ+U∑ini,↑ni,↓0−W/2W/2UupperHubbardbandlowerHubbardbandstrongcorrelationU≫t:reduceddoubleoccupancyc†i↑c†i↓|0ihasenergyUlow-energystatesc†iσ|0isingly-occupied|0iemptysiteslow-energyHamiltoniancanonicaltransformationHt−J≈Heff=eiSHe−iSHeffblock-diagonalindouble-occupancy8canonicaltransformation0−W/2W/2UTdTTT+−ekineticenergytermsT=Te+Td+T++T−H= Te+Td+U∑ini,↑ni,↓+ T++T−perturbationexpansionHeff=eiSHe−iS≈H+i[S,H]+i22[S,[S,H]]+...diagonalindoubleoccupancy⇒S=iU T+−T−+O(t2/U2)projectedHilbert-spaceHt−J=Heff+O(t3/U2)=Te+Td+U∑ini,↑ni,↓+J∑i,jSi·SjJ=4t2/U9Gutzwillerapproximationprojectedwavefunctions|Ψi=P|Ψ0i=∏i(1−ni↑ni↓)|Ψ0iprojectedHilbertspace:|Ψipre-projectedHilbertspace:|Ψ0irenormalizationschemehΨ|ˆO|ΨihΨ|Ψi≈ghΨ0|ˆO|Ψ0ihΨ0|Ψ0irenormalizationfactorsHilbert-spacearguments10Gutzwillerrenormalizationfactors0−W/2W/2UTdTTT+−ekineticenergyhΨ|Te|ΨihΨ|Ψi=hΨ|T|ΨihΨ|Ψi∝nσ(1−n)hΨ0|T|Ψ0ihΨ0|Ψ0i∝nσ(1−nσ)hΨ|T|ΨihΨ|Ψi≈gthΨ0|T|Ψ0ihΨ0|Ψ0igt=1−n1−nσ=1−n1−n/2renorm