1. 强关联电子体系 ― 现象、理论与计算略讲 周森 中国科学院理论物理研究所 2012 年 8 月 9 日 2012 年理论物理研究生暑期学校 凝聚态物理 ― 现象、理论与计算

2. Correlation: motion of one influences the others Uncorrelated: Light traffic = “ideal gas” Correlations and emergence Correlations: jammed (Emergence) Controlled correlations: Fast and efficient Correlations and emergence are everywhere. Correlated particles: electrons, atoms, molecules, grains, biological structures, cars…

3. Biological correlations The human heart is developmentally programmed to occur in the same position again and again.

4. Atomic/molecular correlations Self-assembling of nano-metals in a solution Single crystal: very strong correlations Packing of ions in metallic glasses

5. Electronic correlations 1 cm 3 of matter ~ 10 23 atoms, electrons Weak correlations  conventional material, semiconductor Strong 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: t Coulomb: U Correlation strength: t /U

6. Comparison Conventional Materials Correlated Electron Materials Large overlap of s+p orbitals gives very extended wavefunctions Localization of d+f orbitals enhances Coulomb interaction High quality and flexible fabricationMaterials chemistry challenging! Sensitivity due to weak donor/acceptor binding Sensitivity due to competing ordered states No intrinsic magnetism or other correlations Diverse magnetic and other correlations Intrinsic length scale = large effective Bohr radius a 0 Intrinsic length scales as short as atomic size Weak correlation and large a 0 enable simple and accurate modeling Strong correlations, very challenging to existing theoretical tools Potential gain: new multifunctional materials and devices, which do more and do it better than semiconductors do. Challenges: Understanding phenomena, controlling materials and interfaces

7. Important landmark in superconductivity Sm(O 1-x F x )FeAs Mar. (2008) Conventional SC: 1911 – onwards: Electron-phonon superconductors 1957 – BCS Theory Highest T c : MgB 2 (2001) Unconventional SC: 1977: Heavy Fermion SC 1986: High –T c cuprates (many T c >77K) LaSrCuO, YBaCuO, BiSrCaCuO … 1995: Ruthenates SrRuO, CaSrRuO 2003: Cobaltates NaCoO 2 2008: Fe-Pnictides LaOFeAs, BaFe 2 As 2 … SC: perfect conductor + perfect diamagnetism

8. SC is not driven by electron-phonon interaction – easy to argue (e-ph interactions can still be important) Electron-electron interaction is the driving force – Difficult to prove Unconventional superconductivity “New clues”: unconventional SCs are often found in the vicinity of electronic ordered phases induced by interactions Magnetism : Cuprates, Heavy Fermions, Fe-pnictides Charge order: Organics, transition-metal chalcogenides, Cu x TiSe 2, Ba 1-x K x BaO 3 Electronic Phase separation/Nematic: Cuprates, Fe-pnictides.

9.  What are the most basic properties of a doped Mott insulator? Experimental evidence from the cuprates  How to construct theoretical models for cuprates: from Cu d-electrons and O p-electrons to the effective t-J model  What are the most essential ideas of Resonance Valence Bond? Short-range RVB from the t-J model  Slave-boson formulation High-T c Cuprates

10. Y-123 CuO 2 CuO YBa 2 Cu 3 O 7 La-214 Cuprates: crystal structure  Two-dimensional layered structure  Most important physics in the common CuO 2 plane Bi-2212

11. Octahedral crystal field splitting Cu 2+ [Ar]3d 9 JT distortion O 2- [Be]2p 6 Undoped cuprates have 1e/unit cell → half-filled, low spin S=1/2 Cuprates: electronic configuration p x, p y p z

12. Copper Oxygen 3d electrons 2p electrons Cuprates are p-d charge transfer systems Transition metals Cu, Fe, Co, Ru (4d) … Oxygen, Pnictigen, Chalcogen O, As, P, Se, Te …..  Two-dimensional layered structure  Most important physics in the common CuO 2 plane Cuprates: p-d charge transfer

13. Cuprates: p-d charge transfer insulators Electron Picture: as 1 electron on 3d 8 U = 7 ~ 10 eV Energy cost for charge transfer ∆ pd = U-ε p = 1 ~ 2eV In contrast, Fe-pnictides are charge transfer metals How to describe Cu 2+ =3d 9 ? p-d electron transfer U εpεp 3d 8 3d 10 U U-εpU-εp 3d 8 3d 10 p-d hole transfer Cu OO Hole Picture: as 1 hole on 3d 10

14. Cuprates: doped AF Mott insulator  Everything starts from doping a half-filled AF Mott insulator  This phase diagram is one realization of doping a charge transfer Mott insulator  High-T c is a strong correlation problem. No weak-coupling analog!

15. Cuprates: doped AF Mott insulator Most essential ingredients of a doped Mott insulator: Particle number = 1- x Mobile carrier density/coherence factor = x Experimental evidence Large Fermi surface with Luttinger volume proportional to 1- x (ARPES + Quantum Oscillations)

16. Cuprates: doped AF Mott insulator Quantities having to do with coherent motion of holes scale with doping x (Optical spectroscopy and transport) Padilla et al, PRB (2005)

17. Coherent weight Z A Cuprates: doped AF Mott insulator Low temperature QP coherence weight scales with x (ARPES) Ding et. al., PRL87, 227001 (2001)

18. These essential properties do not come from weakly interacting electrons Strong correlation physics is required to understand cuprates Construction of theoretical models for the strongly correlated CuO 2 plane

19. Strongly correlated electrons on CuO 2 plane Minimal 3-band model in hole picture. Cu: 3d x2-y2 (d), Planar O: 2p x,2p y Example set: t pd = 1 eV, t pp = 0.5 eV, ε p -ε d 0 = 6.5 eV, U = 10 eV This is an Anderson Lattice Model:  Charge transfer insulator when undoped (one hole per Cu) if t pd <  pd =U-(ε p -ε d )  Charge fluctuations quenched by large-U on Cu: only one-hole state allowed → local moment (S=1/2) on Cu. Large quantum spin fluctuations  Doped holes go to oxygen; new states introduced inside the charge transfer gap.

20. Zhang and Rice, PRB37, 3759 (1988).  Doping introduces new states in CT gap: Keeping the lowest one-hole and the lowest two-hole ZR singlet O Cu Operator A dd A 2-hole bound state hopping → effective single-hole hopping.  Single-band t-J model:  Undoped case: Charge-transfer insulator → Copper spins interact via superexchange  Spin-1/2 Heisenberg model Effective 1-band model via formation of ZR singlet

21. Systematic derivation using cell perturbation  Construct “molecular” orbitals centered on Cu  Decompose into equivalent (overlapping) CuO 4 cells: H = H 0 (non-interacting Cells) + H cc (cell-cell interactions)  Solve one-cell problem,  Construct effective Hamiltonian (Rayleigh-Schrödinger expansion) P 0 : projection operator for |  of h 0  Express H in the basis of the one-cell states: Jefferson, Eskes & Feiner, PRB 45, 7959 (1992); Feiner, Jefferson & Raimondi, PRB 53, 8751 (1996); …

22. Simplest description of the cuprates Undoped CuO 2 plane AF Heisenberg model  Dope holes t J t  3J, t’  -0.3t t’ Doped CuO 2 plane One-band t-J model No charge fluctuations P G : projection operator No double occupation Connection to Hubbard model in the large-U limit

23. Basic physics of strong correlation Band (Pauli) Insulator = Even number of e - per site Mott Insulator = Interaction driven insulator half-filled case = one e - per site Hubbard Model Why is it difficult to solve? Single-site Hilbert space: Total number of states : lattice with N s sites: 4 Ns

24. Two-site Hubbard model Consider subspace with n=2, S z =0: 4 states Eigenvalues: Total number of states=4 2 =16, H is a 16  16 matrix. Due to symmetries, electron number n and S z and a good quantum number, H block diagonal for different n and S z

25. Upper and lower Hubbard bands E DOS LHB UHB W  t double occ. 0 -J U U+J UMott-Hubbard Gap Lowest energy state: S=0 Spin singlet bond Two sites

26. Large U: projection of upper Hubbard band Virtual hopping favors AFM correlations For large U>>t, project out doubly occupied sites Canonical transformation → t-J model in projected Hilbert space

27. Many theories for high-Tc, but the theory is … RVB Resonating Valence bond 共振共价键

28. Why doping an AF Mott insulator  SC RVB state for Heisenberg model on triangular lattice (1973) Anderson resonating valence bond (RVB) idea Superposition of spin-singlet pairs rather than Neel state Better account for the quantum fluctuations: spin liquid state RVB picture for high-T c cuprates (1987) Electrons form spin-singlet pairs, mobilized upon doping and condense into a SC state

29. But we know cuprates have antiferromagnetic order at x=0 Hole doping x 0 Frustration … AF T=0 Spin Liquid SC RVB Cuprates Hole doping x 0 AF Hidden Spin Liquid SC T Short-range RVB

30. In RVB picture: There is not a pairing mechanism. Spin singlet pairs already exist. SC comes from coherence of pairs Spin pairing through instantaneous superexchange interaction. How does this picture materialize through t-J model?

31. Spin liquid state on a S=1/2 Kagome lattice Most frustrated 2D lattice

32.  Slave-boson for projected Hilbert space: Spinless boson Spin-carrying fermion (spinon) Each site is and must be occupied by either a boson or a fermion Constraints → equality: Enforced by Lagrange multipliers completeness Slave-boson formulation of the t-J model

33. Slave-boson mean-field theory RVB decoupling of exchange interaction: Paramagnetic valence bond Spin-singlet pairing Uniform mean-field solution: τ ij – symmetry of the pair: s-wave, d-wave, … Not in AFM

34. Doping Concentration x Doping Concentration x T d-SC SG incoherent metal Fermi liquid Kotliar and Liu, PRB38, 5142 (1988). Bose condensationSpinon pairing antinode node Superexchange → d-wave SC PG: spin pairing gap T c is set by phase coherence below optimal doping Uniform mean-field phase diagram