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-Tc cuprates 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
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, …
Z Sentil, Fisher
……..
Variational wave function: Gros, Anderson, Lee, Randeria, Rice, Trivedi,
Zhang; T.K. Lee; Tao Li, …
Lee, Nagaosa, Wen, RMP (2006)

11. Physical meaning of the gauge flux

12. Pseudogap phase T T0 TN T* Tv Tc x antiferromagnetic order
~ J/kB
strange metal: incoherent
Pseudogap:
An RVB state? T0
upper pseudogap phase
strong AF correlations TN
lower pseudogap phase T*
Pseudogap phase: a new state of matter with enormous AF “collective” fluctuations or superconducting phase “collective” fluctuations. Fermi liquid to SDW, CDW, SC transitions usually only involve very narrow critical fluctuation regions. So the pseudogap phase must incorporate the “collective” phenomena intrinsically.
A mean-field description of strong fluctuations? Tv
strong SC fluctuations Tc x
QCP
antiferromagnetic order
d-wave superconducting order

13. PLAN Introduction: High-Tc phenomenology Pseudogap phenomenon
High-Tc cuprates 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

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

15. Bosonic RVB state

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

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

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

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

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

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

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

23. 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

24. 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: + - + - + + + + - - + - - + - - - - + - + + - + +

25. Phase string model - Effective theory
spinon part
holon part
Novel but simple structure
mutual duality/mutual
Chern-Simons gauge structure

26. 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

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

28. 1D case i
Correct correlation functions
Weng, et al. (1997)

29. 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

30. Half-filling Schwinger boson representation RVB condensation T<T0
Many-body, strong correlation, lead to
quantum fluctuations
RVB pairing
High-temperature series expansion

31. Upper pseudogap phase (UPP) at finite doping
Zhengcheng Gu & ZYW, PRB (2005)

32. Spin channel - continued
Zhengcheng Gu & ZYW, PRB (2005)
NMR spin relaxation rates
YBa2Cu3O7-y

33. Charge channel RVB disappears a lot of unpaired spins excited
A. Sanander-Syro, et al.
PRB (2004)
Y.S. Lee et al. PRB 72, (2005)
Zhengcheng Gu & ZYW (2007)

34. Pseudogap phase T T0 TN T* Tv Tc x d-wave superconducting order ~ J/kB
strange metal: incoherent T0
upper pseudogap phase
strong AF correlations
lower pseudogap phase TN T*
strong SC fluctuations
Pseudogap phase: a new state of matter with enormous AF “collective” fluctuations or superconducting phase “collective” fluctuations. Fermi liquid to SDW, CDW, SC transitions usually only involve very narrow critical fluctuation regions. So the pseudogap phase must incorporate the “collective” phenomena intrinsically.
A mean-field description of strong fluctuations? A “liquid” is not easy to describe, including the Mott paramagnet. Tv Tc x
QCP
d-wave superconducting order

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

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

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

38. Spin channel π S=1 excitation 0.05 0.125 0.2
Weiqiang Chen & ZYW, PRB (2005)
S=1 excitation
0.05
0.125
C. Stock, et al. PRB (2004)
0.2

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

40. Pseudogap phase T T0 TN T* Tv Tc x antiferromagnetic order
~ J/kB
strange metal: incoherent
Pseudogap:
Mutual Chern-Simons? T0
upper pseudogap phase
strong AF correlations TN
lower pseudogap phase T*
Pseudogap phase: a new state of matter with enormous AF “collective” fluctuations or superconducting phase “collective” fluctuations. Fermi liquid to SDW, CDW, SC transitions usually only involve very narrow critical fluctuation regions. So the pseudogap phase must incorporate the “collective” phenomena intrinsically.
A mean-field description of strong fluctuations? Tv
strong SC fluctuations Tc x
QCP
antiferromagnetic order
d-wave superconducting order

41. 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

42. Superconducting phase transition
Tc formula

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

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

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

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

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

48. Lattice gauge field theory Peng Ye, C. S. Tian, X. L
Lattice gauge field theory Peng Ye, C.S. Tian, X.L. Qi, and ZYW, arXiv Supeng Kou, X.L. Qi and ZYW, PRB (2005)

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

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

51. Intermediate δ: Bose insulating phase
non-analytic; signaling spinon deconfinement QCP
spinon deconfinement
holon deconfinement
non-analytic; signaling holon deconfinement QCP

52. Straightforward application of composition rule: BI phase is indeed insulating!
antiferrom-agnet
Bose insulator
supercondu-ctor
holon conductivity ∞
spinon conductivity
electric conductivity
dual！

53. Sumary of the phase string approach
~ J/kB
strange metal: incoherent
Electron fractionalization
with emergent mutual
Chern-Simons T0 TN T*
Pseudogap phase: a new state of matter with enormous AF “collective” fluctuations or superconducting phase “collective” fluctuations. Fermi liquid to SDW, CDW, SC transitions usually only involve very narrow critical fluctuation regions. So the pseudogap phase must incorporate the “collective” phenomena intrinsically.
A mean-field description of strong fluctuations? Tv Tc x
QCP
antiferromagnetic order BI
d-wave superconducting order
Low-T instabilities

54. Charge-spin entanglement induced by phase string+ + + - + - + + - + + - - + + + + + + + - + + + +
AFM state
Nagaoka state
an extreme case ignoring
the superexchange energy
Signs are cancelled out in the SC ground state such that all electrons participate in pairing, in contrast to the BCS Cooper pairing for electrons near the Fermi energy. + - - + + +
RVB/Pseudogap - - + + - - + -
minimizing the total exchange
and kinetic energy

55. ? Sumary of the phase string approach T T0 TN T* Tv Tc x
~ J/kB
strange metal: incoherent
Electron fractionalization
with emergent mutual
Chern-Simons T0 TN T*
Pseudogap phase: a new state of matter with enormous AF “collective” fluctuations or superconducting phase “collective” fluctuations. Fermi liquid to SDW, CDW, SC transitions usually only involve very narrow critical fluctuation regions. So the pseudogap phase must incorporate the “collective” phenomena intrinsically.
A mean-field description of strong fluctuations? Tv ? Tc
QP coherence x
QCP
antiferromagnetic order
d-wave superconducting order
Low-T instabilities

56. Mott collapse at QCP? QCP In this slide, emphasize the sign problem:
At half-filling, Marshall sign technique absorbs the sign completely,
However, when holes are doped, the additional sign comes out in H_t,
In slave-particle technique, the sign “\sigma” is related to spin configuration. Furthermore, spin configuration evolves through its own dynamics. The sign becomes notorious!
QCP

57. Summary
Pseudogap holds the key to understanding the (underdoped) high-Tc problem
Doped Mott insulator/antiferromagnet: A sensible starting-point model
Pseudogap = bosonic RVB state
Electron fractionalization with emergent mutual Chern-Simons gauge structure
A self-consistent and systematic account for pseudogap phenomena,
superconductivity and antiferromagnetic instabilities, and the global phase
diagram
Non-Landau paradigm: topological phase transitions: spinon and holon
confinement and deconfinement

58. Collaborators V. N. Muthukumar (CCNY)
Supeng Kou (Beijing Normal Univ.)
Yi Zhou (Zhejiang Univ.)
Weiqiang Chen (Hong Kong Univ.)
Xiao-Liang Qi (Stanford)
Zheng-Cheng Gu (KITP)
Kai Wu (Tsinghua)
Jan Zaanen (Leiden Univ.)
Peng Ye (Tsinghua)
Jiawei Mei (Tsinghua)
Chushun Tian (Köln Univ.)
…….

59. Why strong correlations give rise to high-Tc superconductivity
*superexchange coupling J～1,500 K
RVB at T0
*Why Tc is so low as compared to the pseudogap temperature T0?
doped Mott insulator & phase string effect
How to mathematically describe a strongly correlated system
restricted Hilbert space
sign structure
New state of matter: Pseudogap phase
*hidden order parameter: RVBs, symmetry: internal U(1)
*mutual Chern-Simons gauge theory description (not a MFT)

60. Superconducting phase: low-T instability of the pseudogap phase
*emergent; d-wave symmetry; nodal QP (BCS-like)
*why d-wave is not the whole issuse
*gluon?
Mott physics: emphasis?
Mott gap; Heisenberg antiferromagnet at half-filling
doped Mott insulator: t-J model
Electron fractionalization: non-Landau-paradigm
holon; spinon; confinement-deconfinement
spin-roton; spinon-vortex

61. Thank you
For your attention!