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强关联电子体系 ― 现象、理论与计算略讲

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. 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. Correlations and emergence. Correlation: motion of one influences the others. Uncorrelated: Light traffic = “ideal gas”. (Emergence) Controlled correlations: Fast and efficient.

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强关联电子体系 ― 现象、理论与计算略讲

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  1. 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 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

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

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

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

  5. Electronic correlations 1 cm3 of matter ~ 1023 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 Correlation strength: t/U Coulomb: U

  6. Comparison • 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 SC: perfect conductor + perfect diamagnetism Conventional SC: 1911 – onwards: Electron-phonon superconductors 1957 – BCS Theory Highest Tc: MgB2 (2001) Unconventional SC: 1977: Heavy Fermion SC 1986: High –Tc cuprates (many Tc>77K) LaSrCuO, YBaCuO, BiSrCaCuO … 1995: Ruthenates SrRuO, CaSrRuO 2003: Cobaltates NaCoO2 2008: Fe-Pnictides LaOFeAs, BaFe2As2 … Sm(O1-xFx)FeAs Mar. (2008)

  8. Unconventional superconductivity 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 “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, CuxTiSe2, Ba1-xKxBaO3 Electronic Phase separation/Nematic: Cuprates, Fe-pnictides.

  9. High-Tc Cuprates • 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

  10. Cuprates: crystal structure La-214 Y-123 Bi-2212 CuO2 CuO2 CuO CuO2 CuO2 YBa2Cu3O7 • Two-dimensional layered structure • Most important physics in the common CuO2 plane

  11. Cuprates: electronic configuration Cu2+ [Ar]3d 9 O2- [Be]2p6 px, py pz Octahedral crystal field splitting JT distortion Undoped cuprates have 1e/unit cell → half-filled, low spin S=1/2

  12. Cuprates: p-d charge transfer • Two-dimensional layered structure • Most important physics in the common CuO2 plane Copper 3d electrons Oxygen 2p electrons Cuprates are p-d charge transfer systems Transition metals Cu, Fe, Co, Ru (4d) … Oxygen, Pnictigen, Chalcogen O, As, P, Se, Te …..

  13. Cuprates: p-d charge transfer insulators How to describe Cu2+=3d9 ? 3d10 Electron Picture: as 1 electron on 3d8 p-d electron transfer U εp 3d8 U = 7 ~ 10 eV Energy cost for charge transfer ∆pd= U-εp= 1 ~ 2eV Cu O Cu O 3d8 p-d hole transfer U Hole Picture: as 1 hole on 3d10 U-εp 3d10 In contrast, Fe-pnictides are charge transfer metals

  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-Tc 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. Cuprates: doped AF Mott insulator  Low temperature QP coherence weight scales with x (ARPES) Coherent weight ZA 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 CuO2 plane

  19. Strongly correlated electrons on CuO2 plane Minimal 3-band model in hole picture. Cu: 3dx2-y2(d), Planar O: 2px,2py Example set: tpd = 1 eV, tpp = 0.5 eV, εp-εd0= 6.5 eV, U = 10 eV • This is an Anderson Lattice Model: • Charge transfer insulator when undoped (one hole per Cu) if tpd < pd=U-(εp-εd) • 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. Effective 1-band model via formation of ZR singlet Zhang and Rice, PRB37, 3759 (1988). • Undoped case: • Charge-transfer insulator → Copper spins interact via superexchange  Spin-1/2 Heisenberg model • Doping introduces new states in CT gap: • Keeping the lowest one-hole and the lowest two-hole ZR singlet Ad dA O Cu Operator 2-hole bound state hopping → effective single-hole hopping.  Single-band t-J model:

  21. Systematic derivation using cell perturbation Jefferson, Eskes & Feiner, PRB 45, 7959 (1992); Feiner, Jefferson & Raimondi, PRB 53, 8751 (1996); … Construct “molecular” orbitals centered on Cu  Decompose into equivalent (overlapping) CuO4 cells: Solve one-cell problem, H = H0 (non-interacting Cells) + Hcc (cell-cell interactions)  Construct effective Hamiltonian (Rayleigh-Schrödinger expansion) P0: projection operator for | of h0  Express H in the basis of the one-cell states:

  22. Simplest description of the cuprates Undoped CuO2 plane AF Heisenberg model Doped CuO2 plane One-band t-J model Dope holes t’ t  t 3J, t’ -0.3t J No charge fluctuations PG: 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 Ns sites: 4Ns

  24. Two-site Hubbard model Total number of states=42=16, H is a 1616 matrix. Due to symmetries, electron number n and Sz and a good quantum number, H block diagonal for different n and Sz Consider subspace with n=2, Sz=0: 4 states Eigenvalues:

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

  26. Large U: projection of upper Hubbard band For large U>>t, project out doubly occupied sites Virtual hopping favors AFM correlations 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 Anderson resonating valence bond (RVB) idea RVB state for Heisenberg model on triangular lattice (1973) Superposition of spin-singlet pairs rather than Neel state Better account for the quantum fluctuations: spin liquid state RVB picture for high-Tc 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 Short-range RVB Frustration … Spin Liquid T=0 RVB T Hidden Spin Liquid SC Cuprates AF AF SC 0 Hole doping x 0 Hole doping x

  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 formulation of the t-J model  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 completeness Constraints → equality: Enforced by Lagrange multipliers

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

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

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