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

1. High-Tc superconductivity in doped antiferromagnets (II) Zheng-Yu Weng Institute for Advanced Study Tsinghua University, Beijing KITPC AdSCFT/CM Nov. 5, 2010

2. Outline • Introduction: High-Tc phenomenology Pseudogap phenomenon • High-Tccuprates as doped Mott insulators /doped antiferromagnets • Pseudogap state as an RVB state and the slave-boson approach • Reduced fermion signs in doped Mott insulator: emergent mutual Chern-Simons gauge fields • Conclusion

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

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

5. Gauge structure (half-filling)

6. Mean field theory of fermionic RVB state

7. Effect of the gauge structure

8. Finite doping U(1) symmetry

9. d-wave Baskaran, Zou, Anderson (1988) Zhang,Gros, Rice, Shiba (1988) Kotliar, Liu (1988) … Slave-boson mean-field theory

10. Fermionic RVB theories basic wave function Gutzwiller projection 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, … Z2 Sentil, Fisher …….. Variational wave function: Gros, Anderson, Lee, Randeria, Rice, Trivedi, Zhang; T.K. Lee; Tao Li, … Lee, Nagaosa, Wen, RMP (2006)

11. Pseudogap phase T ~ J/kB strange metal: incoherent Pseudogap: An RVB state? 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

12. PLAN • Introduction: High-Tc phenomenology Pseudogap phenomenon • High-Tccuprates as doped Mott insulators /doped antiferromagnets • Pseudogap state as an RVB state and the slave-boson approach • Reduced fermion signs in doped Mott insulator: Pseudogap - intrinsic mutual Chern-Simons gauge fields • Conclusion

13. Absence of fermion signs at half-filling Mott insulator Heisenberg model Total disapperance of fermion signs!

14. BosonicRVB state

15. T Ground state at half-filling doping X=0 Bosonic RVB wavefunction Liang, Doucot, Anderson, PRL (1988) A spin singlet pair

16. Reduced fermion signs in doped case: single hole case - + + + + - - + - - + + - - + loop c Singular Phase String Effect! Phase String Effect

17. Phase string captured by mutual Chern-Simons fields flux tube anyonsigns Wilczek: Anyons (1982) Chern-Simons gauge field Phase string signs Mutual Chern-Simons (semion) gauge field

18. Exact mapping between the phase string and a mutual Chern-Simons gauge field description - + - Z.Y. Weng, et al PRB (1997); PRL (1998)

19. t-J model in the phase string representation bosonic hopping superexchange RVB order parameter U(1)XU(1) gauge invariance

20. Phase string model - Effective theory spinon part holon part mutual duality/mutual Chern-Simons gauge structure

21. Effective Hamiltonian for the pseudogap phase t-J model: exact sign structure + bosonic RVB Phase string model Emergent new model/physics at mutual Chern-Simons Electron fractionalization

22. One-dimensional case (open BC) no sign problem! phase string spinon holon

23. 1D case i Correct correlation functions Weng, et al. (1997)

24. Effective Hamiltonian for the pseudogap phase t-J model: exact sign structure + bosonic RVB Phase string model Emergent new model/physics at mutual Chern-Simons Electron fractionalization

25. Half-filling Schwinger boson representation RVB condensation T<T0 quantum fluctuations RVB pairing High-temperature series expansion

26. Upper pseudogap phase (UPP) at finite doping ZhengchengGu & ZYW, PRB (2005)

27. Spinchannel - continued ZhengchengGu & ZYW, PRB (2005) NMR spin relaxation rates YBa2Cu3O7-y

28. Chargechannel RVB disappears a lot of unpaired spins excited A. Sanander-Syro, et al. PRB (2004) Y.S. Lee et al. PRB 72, (2005) ZhengchengGu & ZYW (2007)

29. Pseudogap phase T ~ J/kB strange metal: incoherent T0 upper pseudogap phase strong AF correlations lower pseudogap phase TN T* strong SC fluctuations Tv Tc x QCP d-wave superconducting order

30. electromagnetic field Lower pseudogap phase (LPP) π -π Bose condensation Generalized Ginzburg-Landau equaion

31. B v -T Spinon vortices Nernst effect Xu et al., Nature (2000), Wang et al., PRB (2001). Xiaoliang Qi, ZYW (2006)

32. Mutual Duality π-flux π-flux duality spinon chargon

33. Spinchannel π WeiqiangChen & ZYW, PRB (2005) S=1 excitation 0 0.05 0.125 C. Stock, et al. PRB (2004) 0.2

34. Lower pseudogap phase (LPP) Z.C. Gu and Z. Y. Weng (2007)

35. Pseudogap phase T ~ J/kB strange metal: incoherent Pseudogap: Mutual Chern-Simons? 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

36. Spinon confinement in SC The low-lying excitations are degenerate Spin-Rotons as paired spinon-vortices: S=1 S=0 Jiawei, ZYW (2009) Spin-Charge entanglement

37. Superconducting phase transition Tc formula

38. “resonant mode” in neutron scattering Raman scattering in A1g channel

39. Superconducting transition spinon confinement-deconfinement transition Spin-rotons new elementary excitations in SC phase

40. Predictions spinon-vortex magnetic vortex with free moment Zn impurity dissipationless conserved spin Hall effect

41. Zinc impurity single spinon Xiaoliang Qi and ZYW (2005)

42. Probing single spinon: S=1/2 at a magnetic vortex core single spinon-vortex Muthukumar & ZYW (2002) W. Halperin (2003)

43. Lattice gauge field theoryPeng Ye, C.S. Tian, X.L. Qi, and ZYW, arXiv1007.2507Supeng Kou, X.L. Qi and ZYW, PRB (2005)

44. Nonlocal order parameters probing confinement - deconfinement of holons spinons

45. Quantitative results of two limits: dual! antiferromagnetic phase superconducting phase holon confinement spinon confinement

46. Intermediate δ: Bose insulating phase non-analytic; signaling spinondeconfinement QCP spinondeconfinement holondeconfinement non-analytic; signaling holondeconfinement QCP

47. Straightforward application of composition rule: BI phase is indeed insulating! dual！

48. Sumary of the phase string approach T ~ J/kB strange metal: incoherent Electron fractionalization with emergent mutual Chern-Simons T0 TN T* Tv Tc x QCP antiferromagnetic order BI d-wave superconducting order Low-T instabilities

49. 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 + - -

50. Sumary of the phase string approach T ~ J/kB strange metal: incoherent Electron fractionalization with emergent mutual Chern-Simons T0 TN T* Tv ? Tc QP coherence x QCP antiferromagnetic order d-wave superconducting order Low-T instabilities