1. Zheng-Yu Weng Institute for Advanced Study Tsinghua University, Beijing Newton Institute, Cambridge 2013.9.16 Mott Physics, Sign Structure, and High-Tc Superconductivity

2. Outline Introduction to basic experimental phenomenology of high-T c cuprates High-T c cuprates as doped Mott insulators /doped antiferromagnets Basic principles: Mott physics and sign structure Nontrivial examples: (1) one-hole case (2) finite doping and global phase diagram (3) ground state wavefunction Summary and conclusion

3. MuellerBednorz Discovery of high-T c superconductors 1986

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

5. La 2-x Sr x CuO 4 Spin susceptibility (T. Nakano, et al. (1994)) Specific heat (Loram et al. 2001) NMR spin-lattice relaxation rate (T. Imai et al. (1993)) Pauli susceptibility Korringa behavior Sommerfeld constant Fermi liquid behavior:

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

7. Photoemission Optical measurement NMR 1/T 1 Nernst effect uniform susceptibility, resistivity

8. d-wave superconducting order T T0T0 0 antiferromagnetic order ~ J/k B strong SC fluctuations strong AF correlations lower pseudogap phase Underdoped phase diagram strange metal: maximal scattering T*T* TNTN TvTv TcTc QCP Pseudogap: New quantum state of matter? A non-Fermi-liquid x FL

9. Outline Introduction to basic experimental phenomenology of high-T c cuprates High-T c cuprates as doped Mott insulators /doped antiferromagnets Basic principles: Mott physics and sign structure Nontrivial examples: (1) one-hole case (2) finite doping and global phase diagram (3) ground state wavefunction Summary and conclusion

10. cuprates iron pnictides

11. T T0T0 x ~ J/k B T*T* TNTN TvTv TcTc QCP Half-filling: Mott insulator x=0 Anderson, Science 1987 Cuprates = doped Mott Insulator one-band large-U Hubbard model:

12. Mott Insulator/ antiferromagnet Mott insulator doped Mott insulator Heisenberg model t-J model

13. hopping superexchange A minimal model for doped Mott insulators: t-J model

14. Pure CuO 2 plane H = J S i · S j large J = 135 meV quantum spin S =1/2 Half-filling: Low-energy physics is described by Heisenberg model

15. Ando et al, PRL 87, 017001 (2001) K. M. Shen et al, PRL 93, 267002 (2004) ARPES result: A broad peak at x=0 charge localization at low doping

16. Sebastian, et al., Reports on progress in physics 75, 102501 (2012) La-Bi2201 Peng, et al., arXiv:1302.3017 (2013) La-Sr-Cu-O Doping the Mott Insulator/ antiferromagnet

17. Sebastian, et al., Reports on progress in physics 75, 102501 (2012) charge localization La-Bi2201 Peng, et al., arXiv:1302.3017 (2013) La-Sr-Cu-O Doping the Mott Insulator/ antiferromagnet

18. If charge localization is intrinsic in a doped Mott insulator with AFLRO? If charge delocalization (superconductivity) arises by destroying the AFLRO? Is localization-delocalization the underlying driving force or the T=0 phase diagram of the underdoped cuprates? Questions

19. Outline Introduction to basic experimental phenomenology of high- T c cuprates and high-T c cuprates as doped Mott insulators /doped antiferromagnets Basic principles: Mott physics and sign structure Nontrivial examples: (1) one-hole case (2) finite doping and global phase diagram (3) ground state wavefunction Summary and conclusion

20. Statistical sign structure for Fermion systems Fermion signs Landau Fermi Liquid

21. nodal hypersurface Nodal hypersurface Pauli hypersurface Test particle d=2 interacting fermions: fractal nodes F. Kruger and J. Zaanen, (2008)

22. (1)Fermi liquid: Fermion signs (2)Off Diagonal Long Rang Order (ODLRO): compensating the Fermion signs Bose condensation Cooper pairing in SC state CDW (“exciton” condensation) SDW (weak coupling) normal state: Fermi liquid Antiferromagnetic order (strong coupling) Complete disappearance of Fermion signs!

23. Phase string effect D.N. Sheng, Y.C. Chen, ZYW, PRL (1996) (3) Single-hole doped Heiserberg model: ＋ －

24. at arbitrary doping, dimensions, temperature Wu, Weng, Zaanen, PRB (2008) = total steps of hole hoppings = total number of spin exchange processes = total number of opposite spin encounters (4) Exact sign structure of the t-J model

25. + - + + - + + + + + + + + + - - - - - - - - - - + For a given path c: (-) (-) 3 K. Wu, ZYW, J. Zaanen, PRB (2008)

26. C. N. Yang (1974), Wu and Yang (1975) A B Nonintegrable phase factor: Emergent gauge force in doped Mott insulators! “An intrinsic and complete description of electromagnetism” “Gauge symmetry dictates the form of the fundamental forces in nature” Mutual Chern-Simons gauge theory ZYW et al (1997) (1998) Kou, Qi, ZYW PRB (2005); Ye, Tian, Qi, ZYW, PRL (2011); Nucl. Phys. B (2012)

27. “smooth” paths good for mean-field treatment singular quantum phase interference Mott physics = phase string sign structure replacing the Fermion signs Strong correlations = charge and spin are long-range entangled Sign structure + restricted Hilbert space = unique fractionalization New guiding principles:

28. Outline Introduction to basic experimental phenomenology of high- T c cuprates and high-T c cuprates as doped Mott insulators /doped antiferromagnets Basic principles: Sign structure and Mott physics Nontrivial examples: (1) one-hole case (2) finite doping and global phase diagram (3) ground state wavefunction Summary and conclusion

29. DMRG numerical study t-J ladder systems Z. Zhu, H-C Jiang, Y. Qi, C.S. Tian, ZYW, Scientific Report 3, 2586 (2013 ）

30. Effect of phase string effect σ no phase string effect Self-localization of the hole!

31. σ Removing the phase string: A sign-free model no phase string effect!

32. Momentum distribution without phase string effect Quasiparticle picture restored!

33. t’ t localization-delocalization transition

34. - - + + - - - + + + + - + D.N. Sheng, et al. PRL (1996); ZYW, et al. PRB (2001) Theoretical understading of self-localization of the one-hole in 2D - Holon localization at low doping: S.P. Kou, ZYW, PRL (2003) T.-P. Choy and Philip Phillips, PRL (2005) P. Ye and Q.R. Wang, Nucl. Phys. B (2013) destructive quantum phase interference leads to self-localization

35. Outline Introduction to basic experimental phenomenology of high- T c cuprates and high-T c cuprates as doped Mott insulators /doped antiferromagnets Basic principles: Sign structure and Mott physics Nontrivial examples: (1) one-hole case (2) finite doping and global phase diagram (3) ground state wavefunction Summary and conclusion

36. Example II: Delocalization and superconductivity - - + + - - - + + + + - + - - - + + - - - + + + - - + localization/AFLROdelocalization/SC spin liquid/RVB! AF spin liquid doping SC localization

37. - - + + - - - + + + - - + Non-BCS elementary excitation in SC state - - + + - - + + - - + - - + + - + - - + + - Superconducting transition spin-roton spinon-vortex spinon confinement-deconfinement transition

38. T T0T0 δ AF SC FL pseudogap AF = long-range RVB localization “strange metal” Global phase diagram charge-spin long-range entanglement by phase string effect

39. Outline Introduction to basic experimental phenomenology of high- T c cuprates and high-T c cuprates as doped Mott insulators /doped antiferromagnets Basic principles: Sign structure and Mott physics Nontrivial examples: (1) one-hole case (2) finite doping and global phase diagram (3) ground state wavefunction Summary and conclusion

40. Example III : “Parent” ground state jdjd lhlh iuiu Superconducting state: emergent (ghost) spin liquid AFM state: ZYW, New J. Phys. (2011) short-ranged

41. Electron fractionalization form

42. Cuprates are doped Mott insulators with strong Coulomb interaction New organizing principles of Mott physics: An altered fermion sign structure due to large-U Consequences: (1) Intrinsic charge localization in a lightly doped antiferromagnet (2) Charge delocalization (superconductivity) arises by destroying the AFLRO (3) Localization-delocalization is the underlying driving force for the T=0 phase diagram of the underdoped cuprates Non-BCS-like ground state wavefunction Summary and Conclusion

44. P. W. Anderson: Resonating valence bond (RVB) theory (1987) Slave-boson mean-field theory: Baskaran, Zou, Anderson (1988) Kotliar, Liu (1988) … Gauge theory description: U(1) P.A. Lee, N. Nagaosa, A. Larkin, … SU(2) X.G. Wen, P. A. Lee, … Z 2 Sentil, Fisher …….. Variational wave function: Gros, Anderson, Lee, Randeria, Rice, Trivedi, Zhang; T.K. Lee; Tao Li, … Fermionic RVB theories Lee, Nagaosa, Wen, RMP (2006) Anderson, et al., J. Phys.: Condens. Mater (2004)

45. (5) Hubbard model on bipartite lattices: A general sign structure (Long Zhang & ZYW, 2013 ) Hilbert space: spinons holon (h) doublon (d) Basic hopping processes in the Hubbard model

46. Partition function ： t UJ + + - + + + - + - + - （-）（-） half-filling:

47. Spin-charge separation three-leg ladder: