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

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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

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MuellerBednorz Discovery of high-T c superconductors 1986

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

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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:

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

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

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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

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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

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cuprates iron pnictides

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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:

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

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

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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

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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

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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

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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

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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

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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

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Statistical sign structure for Fermion systems Fermion signs Landau Fermi Liquid

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

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(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!

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

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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

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

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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)

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“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:

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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

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

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

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

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Momentum distribution without phase string effect Quasiparticle picture restored!

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t’ t localization-delocalization transition

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- - + + - - - + + + + - + 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

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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

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

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

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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

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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

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

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Electron fractionalization form

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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

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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)

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(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

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

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Spin-charge separation three-leg ladder:

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