Zheng-Yu Weng Institute for Advanced Study Tsinghua University, Beijing

1. High-Tc superconductivity in doped antiferromagnets (I) Zheng-Yu Weng Institute for Advanced Study Tsinghua University, Beijing KITPC, AdS/CM duality Nov. 4, 2010

2. Outline • Introduction: High-Tc experimental phenomenology pseudogap phenomenon • High-Tccuprates as doped Mott insulators /doped antiferromagnets exact sign structure • Pseudogap state as an RVB state and the slave-boson approach electron fractionalization and gauge degrees of freedom • Reduced fermion signs in doped Mott insulator: pseudogap - emergent mutual Chern-Simons gauge fields • Conclusion

3. High-Tccuprate superconductors Mueller Bednorz

4. Scientists: dreaming about instant fame Megahype 1990 March meeting: 30 sessions in parallel! over 100000 papers Woodstock of physics Business people: getting rich!

5. Nature of superconducting state? What is the essential elementary excitation deciding the superconducting transition? sharp Bogoliubov QP peak (laser ARPES, XJ Zhou, et al.)

6. BCS theory for superconductivity electron pairing by “glueon”: phonon, AF fluctuations, … Fermi sea -- coupling constant -- Coulomb pseudopotential -- characteristic energy of the glueon Strong coupling theory Pb: Tc=7.19 K λ=1.55, μ*=0.13, ω0=4.8 meV Nb3Ge: Tc=21.2 K λ=1.73, μ*=0.12, ω0=10.7 meV High-Tccuprates: Tc ~ 160 K FeAsbased superconductors: Tc ~56 K typical energy scales:

7. Phase diagram of cuprate superconductors New state of matter? (non-Fermi-liquid) sharp Bogoliubov QP peak (laser ARPES, XJ Zhou, et al.)

8. Landau paradigm ARPES Fermi sea Fermi surface of copper typical Fermi liquid behavior: Fermi degenerate temperature Sommerfeld constant Pauli susceptibility Korringa behavior

9. Landau’s Fermi-liquid: state of interacting electron system in metals = Fermi gas of quasiparticles. Paradigm in crisis Quasiparticle: Fermion with S =1/2, momentum k, energy E(k) QP Fermi surface:

10. La2-xSrxCuO4 Spin susceptibility (T. Nakano, et al. (1994)) Specific heat (Loram et al. 2001) typical Fermi liquid behavior: Sommerfeld constant Pauli susceptibility Korringa behavior NMR spin-lattice relaxation rate (T. Imai et al. (1993))

11. Uniform spin susceptibility Fermi liquid Heisenberg model no indication of Pauli susc. T. Nakano, et al. PRB49, 16000(1994) J

12. Resistivity measurement T. Shibauchi, et al. (2001) T. Nakano, et al. PRB49, 16000(1994)

13. T. Imai et al., PRL 70 (1993) Guo-qing Zheng et al. PRL (2005) Kawasaki, et al. PRL (2010)

14. NMR 1/T1 L. Taillefer- arXiv 1003.2972 Optical measurement Nernst effect Photoemission Y.S. Lee et al. PRB 72, (2005)

15. B v -T Vortex Nernst effect and diamagnetism in the pseudogap regime Xu et al., Nature (2000), Wang et al., PRB (2001).

16. Uemura’s Plot: BEC? Phase fluctuations nodal quasiparticle excitations Emery & Kivelson, (1995) P.A. Lee and X.G. Wen (1997)

17. “resonant mode” in neutron exp. P.C. Dai et al, 2007 “resonant mode” in neutron scattering

18. Raman scattering experiment Sacuto& Bourges’ Group, 2002 Raman scattering in A1g channel

19. Two sets of experiments

20. Pseudogap phase T ~ J/kB strange metal: maximal scattering Pseudogap: New quantum state of matter A non-Fermi-liquid T0 upper pseudogap phase strong AF correlations TN lower pseudogap phase T* Tv strong SC fluctuations Tc x QCP antiferromagnetic order d-wave superconducting order

21. Cuprates = doped Mott Insulator T Anderson, Science 1987 ~ J/kB T0 TN T* Tv Half-filling: Mott insulator Tc x=0 QCP x

22. Half-filling: Mott Insulator/Heisenberg antiferromagnet Mott insulator Heisenberg model H = JSi·Sj on-site Coulomb repulsion U causes a Mott insulator

23. Pure CuO2 plane H = JSi·Sj nn large J = 135 meV quantum spin S =1/2 Half-filling: Low-energy physics is described by Heisenberg model

24. neutron scattering Raman scattering Spin flip breaks 6 bonds, costs 3J. J～135 meV

25. Antiferromagnetism at x=0 is well described by the Heisenberg model inverse spin-spin correlation length Chakaravarty, Halperin, Nelson PRL (1988)

26. Heisenberg model J: superexchange coupling high-T expansion Mott insulator

27. Resonating Valence Bond (RVB) … + + ≡ RVB pair P. W. Anderson, Science, 235, 1196 (1987)

28. Ground state at half-filling A spin singlet pair Liang, Doucot, Anderson, PRL (1988) Bosonic RVB wavefunction Good understanding of the Mott antiferromagnet/paramagnet at half-filling!

29. Cuprates = doped Mott Insulator T ~ J/kB T0 TN T* Tv Half-filling: Mott insulator Tc x=0 QCP x

30. Doping the Mott Insulator/ antiferromagnet Mott insulator doped Mott insulator t-J model Heisenberg model

31. Single band Hubbard model, or its strong coupling limit, the t-J model Pure CuO2 plane Dope holes t  J t 3 J H = JSi·Sj nn The cuprates are doped Mott insulators no double occupancy constraint

32. A minimal model for doped Mott insulators: t-J model hoppingsuperexchange

33. Mottness and intrinsic guage invariance Conservations of spin and charge separately: Spin-charge separation and emergent gauge fields in low-energy action !

34. Fermi sea Fermion signs Antisymmetry of wave function ARPES Fermi surface of copper Landau-Fermi liquid behavior Sommerfeld coefficient Pauli susceptibility Korringa behavior

35. Fermion signs in Feynman‘s path-integral Imaginary time path-integral formulation of partition function: Fermion signs

36. Absence of fermion signs at half-filling Mott insulator A complete basis such that Marshall sign rule Heisenberg model Total disapperance of fermion signs!

37. Ground state at half-filling A spin singlet pair Liang, Doucot, Anderson, PRL (1988) Bosonic RVB wavefunction Disappearance of the fermion signs at half-filling

38. Reduced fermion signs in doped case: single hole case Phase String Effect - + - - + + + + - - + + - D. N. Sheng, et al. PRL (1996) K.Wu, ZYW, J. Zaanen (2008) - + loop c Phase String Effect

39. Exact sign structure of the t-J model Exact phase string effect in the t-J model arbitrary doping, temperature dimenions = total steps of hole hoppings = total number of spin exchange processes - number of hole loops For a given path C: + - - + + + + + - - + - - + - - - - - + + + - + +

40. Single-particle propagator ＋ － － ＋ ＋ ＋ － － ＋ Phase string factor ＋ － － － －

41. Goldstone theorem for the ground state energy phase string factor

42. Cuprates as doped Mott insulators Overdoping: Recovering more fermion signs Mott insulator: No fermion signs Doped Mott insulator: Reduced fermion signs

43. Charge-spin entanglement induced by phase string - + + + + - - + - + + + - - + + + + + + + + - + + + AFM state Nagaoka state an extreme case ignoring the superexchange energy + - - + + + RVB/Pseudogap - - + + - minimizing the total exchange and kinetic energy + - -

44. Summary • Pseudogap state is firmly established by experiment • as one of the most exotic phases in the cuprates • which is closely related to high-Tc superconductivity • Doped Mott insulator/antiferromagnet provides a suitable • microscopic model to understand the pseudogap physics • Mott constraint leads to a new sign structure greatly • reduced from the fermion signs at low doping

45. Thank you For your attention!