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

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

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

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

Gauge structure (half-filling)

Mean field theory of fermionic RVB state

Effect of the gauge structure

Finite doping U(1) symmetry

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

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)

Physical meaning of the gauge flux

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

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

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

Bosonic RVB state

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

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

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

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

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

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

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

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

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: + - + - + + + + - - + - - + - - - - + - + + - + +

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

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

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

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

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

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

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

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

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)

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

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

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

Mutual Duality π-flux π-flux duality spinon chargon

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

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

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

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

Superconducting phase transition Tc formula

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

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

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

Zinc impurity single spinon Xiaoliang Qi and ZYW (2005)

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

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, arXiv1007.2507 Supeng Kou, X.L. Qi and ZYW, PRB (2005)

Nonlocal order parameters probing confinement - deconfinement of holons spinons

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

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

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

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

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

? 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

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

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

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.) …….

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)

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

Thank you For your attention!