凝聚态物理现象理论与计算I 周森

1、会计学1凝聚态物理现象理论与计算凝聚态物理现象理论与计算I 周森周森Correlation: motion of one influences the othersUncorrelated:Light traffic = “ideal gas”Correlations: jammed(Emergence) Controlledcorrelations:Fast and efficientCorrelations and emergence are everywhere. Correlated particles: electrons, atoms, molecules, grains, bio

2、logical structures, carsThe human heart is developmentally programmed to occur in the same position again and again. Self-assembling of nano-metals in a solutionSingle crystal: very strong correlationsPacking of ions in metallic glasses1 cm3 of matter 1023 atoms, electronsWeak correlations conventio

3、nal material, semiconductorStrong correlations unique materials/device properties Superconductivity Magnetism Spin-charge coupling, e.g. multiferroics Spin-orbital coupling, e.g. topological insulator Large thermopower Controlled many-electron coherence in nanostructures Overlap: tCoulomb: UCorrelat

4、ion strength: t/ULarge overlap of s+p orbitals gives very extended wavefunctionsLocalization of d+f orbitals enhances Coulomb interactionHigh quality and flexible fabricationMaterials chemistry challenging! Sensitivity due to weak donor/acceptor bindingSensitivity due to competing ordered statesNo i

5、ntrinsic magnetism or other correlationsDiverse magnetic and other correlationsIntrinsic length scale = large effective Bohr radius a0Intrinsic length scales as short as atomic sizeWeak correlation and large a0 enable simple and accurate modelingStrong correlations, very challenging to existing theo

6、retical toolsPotential gain: new multifunctional materials and devices, which do more and do it better than semiconductors do.Challenges: Understanding phenomena, controlling materials and interfacesSm(O1-xFx)FeAs Mar. (2008) Conventional SC:1911 onwards: Electron-phonon superconductors1957 BCS Theo

7、ry Highest Tc: MgB2 (2001)Unconventional SC:1977: Heavy Fermion SC1986: High Tc cuprates (many Tc77K) LaSrCuO, YBaCuO, BiSrCaCuO 1995: Ruthenates SrRuO, CaSrRuO2003: Cobaltates NaCoO22008: Fe-Pnictides LaOFeAs, BaFe2As2 SC: perfect conductor + perfect diamagnetismSC is not driven by electron-phonon

8、interaction easy to argue (e-ph interactions can still be important)Electron-electron interaction is the driving force Difficult to prove“New clues”: unconventional SCs are often found in the vicinity of electronic ordered phases induced by interactionsMagnetism : Cuprates, Heavy Fermions, Fe-pnicti

9、desCharge order: Organics, transition-metal chalcogenides, CuxTiSe2, Ba1-xKxBaO3Electronic Phase separation/Nematic: Cuprates, Fe-pnictides.q What are the most basic properties of a doped Mott insulator? Experimental evidence from the cupratesq How to construct theoretical models for cuprates: from

10、Cu d-electrons and O p-electrons to the effective t-J modelq What are the most essential ideas of Resonance Valence Bond? Short-range RVB from the t-J modelq Slave-boson formulationY-123CuO2CuO2CuO2CuO2CuO YBa2Cu3O7La-214q Two-dimensional layered structureq Most important physics in the common CuO2

11、planeBi-2212 Octahedral crystal field splittingCu2+Ar3d 9JT distortionO2-Be2p6Undoped cuprates have 1e/unit cell half-filled, low spin S=1/2px, pypzCopperOxygen3d electrons2p electronsCuprates are p-d charge transfer systemsTransition metalsCu, Fe, Co, Ru (4d) Oxygen, Pnictigen, ChalcogenO, As, P, S

12、e, Te .q Two-dimensional layered structureq Most important physics in the common CuO2 planeElectron Picture: as 1 electron on 3d8U = 7 10 eVEnergy cost for charge transferpd= U-p= 1 2eV In contrast, Fe-pnictides are charge transfer metalsHow to describe Cu2+=3d9 ?p-d electron transferUp3d83d10UU-p3d

13、83d10p-d hole transferCuCuOOHole Picture: as 1 hole on 3d10q Everything starts from doping a half-filled AF Mott insulatorqThis phase diagram is one realization of doping a charge transfer Mott insulatorq High-Tc is a strong correlation problem. No weak-coupling analog!Most essential ingredients of

14、a doped Mott insulator: Particle number = 1-x Mobile carrier density/coherence factor = xExperimental evidence Large Fermi surface with Luttinger volume proportional to 1-x (ARPES + Quantum Oscillations) Quantities having to do with coherent motion of holes scale with doping x (Optical spectroscopy

15、and transport)Padilla et al, PRB (2005) Effective carrier density Drude weight Superfluid density effNxxxCoherentweight ZA Low temperature QP coherence weight scales with x (ARPES)Ding et. al., PRL87, 227001 (2001)These essential properties do not come from weaklyinteracting electrons Strong correla

16、tion physics is required to understand cupratesConstruction of theoretical models for the stronglycorrelated CuO2 planeMinimal 3-band model in hole picture.Cu: 3dx2-y2(d), Planar O: 2px,2py, , , , , , , ,0, H.c. H.c.pdpdppddiiiiiiiiiii xi yixiyii xiyi yixi yiyiHppddn ndpppUttppppppp Example set: tpd

17、 = 1 eV, tpp = 0.5 eV, p-d0= 6.5 eV, U = 10 eVThis is an Anderson Lattice Model: q Charge transfer insulator when undoped (one hole per Cu) if tpd t, project out doubly occupied sitesCanonical transformation t-J model in projected Hilbert space ,14 (. .)t Jiiiji ji jijGGHP c chc PntnJS S24AF superex

18、change: 0No double occ. constraint:=0,1 iicJctU0 Many theories for high-Tc, but the theory is RVBResonating Valence bond共振共价键共振共价键RVB state for Heisenberg model on triangular lattice (1973)Anderson resonating valence bond (RVB) ideaSuperposition of spin-singlet pairs rather than Neel state Better ac

19、count for the quantum fluctuations: spin liquid stateRVB picture for high-Tc cuprates (1987)12 Electrons form spin-singlet pairs, mobilized upon doping and condense into a SC stateBut we know cuprates have antiferromagnetic order at x=0AFRVBRVBAFEE0Frustration AFT=0Spin LiquidSCRVBCuprates0AFHiddenS

20、pin LiquidSCTShort-range RVBIn RVB picture: There is not a pairing mechanism. Spin singlet pairs already exist. SC comes from coherence of pairsSpin pairing through instantaneous superexchange interaction.How does this picture materialize through t-J model?Most frustrated2D latticeempty site: empty0

21、ibsingly occupied site: emptyiif bsingly occupied site: emptyiif bq Slave-boson for projected Hilbert space:iiifcbSpinless bosonSpin-carrying fermion (spinon),1H.c.iiiiiiffj i ijijiji jiffbHt b b ffJbSS,H.constraintc. ()s 1tJijijiii jijHtc cJc c S S0 Each site is and must be occupied by either a bos

22、on or a fermionConstraints equality:1iiiiffb bEnforced by Lagrange multiplierscompleteness22*11()h.c.+(). 44ffijiijijjijiijijjSSfff ff fRVB decoupling of exchange interaction: ijijijijijfff ff fParamagnetic valence bondSpin-singlet pairingiiibbb0 ijijijiibbUniform mean-field solution:ij symmetry of

23、the pair: s-wave, d-wave, Not in AFMTd-SC0,0b SG00b incoherent metal0,0b Fermi liquid0,0b Kotliar and Liu, PRB38, 5142 (1988).Bose condensationSpinon pairingantinodenode Superexchange d-wave SC PG: spin pairing gap Tc is set by phase coherence below optimal dopingThe human heart is developmentally p

24、rogrammed to occur in the same position again and again. 1 cm3 of matter 1023 atoms, electronsWeak correlations conventional material, semiconductorStrong correlations unique materials/device properties Superconductivity Magnetism Spin-charge coupling, e.g. multiferroics Spin-orbital coupling, e.g. topological insulator Large thermopower Controlled many-electron coherence in nanostructures Overlap: tCoulomb: UCorrelation strength: t/UBand (Pauli) Insulator = Even number of e- per siteMott Insulator = Interaction driven insulatorhalf-filled case = one e- per site hopping