Zheng-Yu Weng Institute for Advanced Study Tsinghua University, Beijing KITPC, AdS/CM duality Nov. 4, 2010 High-T c superconductivity in doped antiferromagnets (I)

Outline Introduction: High-T c experimental phenomenology pseudogap phenomenon High-T c cuprates as doped Mott insulators /doped antiferromagnets exact sign structure Pseudogap state as an RVB state and the slave-boson approach electron fractionalization and gauge degrees of freedom Reduced fermion signs in doped Mott insulator: pseudogap - emergent mutual Chern-Simons gauge fields Conclusion

MuellerBednorz High-T c cuprate superconductors

Megahype Scientists: dreaming about instant fame Woodstock of physics Business people: getting rich! over papers 1990 March meeting: 30 sessions in parallel!

Nature of superconducting state? What is the essential elementary excitation deciding the superconducting transition? sharp Bogoliubov QP peak (laser ARPES, XJ Zhou, et al.)

BCS theory for superconductivity electron pairing by “glueon”: phonon, AF fluctuations, … Pb: T c =7.19 K λ=1.55, μ*=0.13, ω 0 =4.8 meV Nb 3 Ge: T c =21.2 K λ=1.73, μ*=0.12, ω 0 =10.7 meV Strong coupling theory -- Coulomb pseudopotential High-T c cuprates: T c ~ 160 K FeAs based superconductors: T c ~56 K -- coupling constant -- characteristic energy of the glueon Fermi sea typical energy scales:

Phase diagram of cuprate superconductors New state of matter? (non-Fermi-liquid) sharp Bogoliubov QP peak (laser ARPES, XJ Zhou, et al.)

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

Paradigm in crisis Landau’s Fermi-liquid: state of interacting electron system in metals = Fermi gas of quasiparticles. Quasiparticle:Fermion with S =1/2, momentum k, energy E(k) QP Fermi surface:

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 typical Fermi liquid behavior:

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

T. Nakano, et al. PRB49, 16000(1994) Resistivity measurement T. Shibauchi, et al. (2001)

Guo-qing Zheng et al. PRL (2005) T. Imai et al., PRL 70 (1993) Kawasaki, et al. PRL (2010)

L. Taillefer- arXiv Photoemission Y.S. Lee et al. PRB 72, (2005) Optical measurement NMR 1/T 1 Nernst effect

Xu et al., Nature (2000), Wang et al., PRB (2001). B v -T-T Vortex Nernst effect and diamagnetism in the pseudogap regime

Uemura’s Plot: BEC? Emery & Kivelson, (1995) P.A. Lee and X.G. Wen (1997) Phase fluctuations nodal quasiparticle excitations

“resonant mode” in neutron exp. “resonant mode” in neutron scattering P.C. Dai et al, 2007

Raman scattering experiment Sacuto& Bourges’ Group, 2002 Raman scattering in A 1g channel

Two sets of experiments

Two soft modes in spin and charge channels from the stripe state?

heavy fermionorganic metal cuprates iron pnictides CDW Are the cuprates any special? Landau vs. non-Landau paradiagm

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

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

Half-filling: Mott Insulator/Heisenberg antiferromagnet Mott insulator Heisenberg model H = J  S i · S j on-site Coulomb repulsion U causes a Mott insulator

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

neutron scattering Raman scattering Spin flip breaks 6 bonds, costs 3J. J ～ 135 meV

Chakaravarty, Halperin, Nelson PRL (1988) inverse spin-spin correlation length Antiferromagnetism at x=0 is well described by the Heisenberg model

high-T expansion J: superexchange coupling Mott insulator

Resonating Valence Bond (RVB) + + ≡ P. W. Anderson, Science, 235, 1196 (1987) RVB pair …

Bosonic RVB wavefunction Liang, Doucot, Anderson, PRL (1988) Ground state at half-filling A spin singlet pair Good understanding of the Mott antiferromagnet/paramagnet at half-filling!

T T0T0 x ~ J/k B T*T* TNTN TvTv TcTc QCP Half-filling: Mott insulator x=0 Cuprates = doped Mott Insulator

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

The cuprates are doped Mott insulators Pure CuO 2 plane Single band Hubbard model, or its strong coupling limit, the t-J model  Dope holes t J t  3 J H = J  S i · S j nn no double occupancy constraint

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

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

Fermion signs Antisymmetry of wave function Fermi sea ARPES Pauli susceptibility Korringa behavior Sommerfeld coefficient Landau-Fermi liquid behavior Fermi surface of copper

Fermion signs in Feynman‘s path-integral Imaginary time path-integral formulation of partition function: Fermion signs

Absence of fermion signs at half-filling Mott insulator Heisenberg model A complete basissuch that Total disapperance of fermion signs! Marshall sign rule

Bosonic RVB wavefunction Liang, Doucot, Anderson, PRL (1988) Ground state at half-filling A spin singlet pair Disappearance of the fermion signs at half-filling

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

= total steps of hole hoppings = total number of spin exchange processes For a given path C: - number of hole loops Exact phase string effect in the t-J model Exact sign structure of the t-J model arbitrary doping, temperature dimenions

Phase string factor ＋ － － ＋ － － － ＋ ＋ － ＋ － － ＋ Single-particle propagator

Goldstone theorem for the ground state energy phase string factor

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

Nagaoka state an extreme case ignoring the superexchange energy RVB/Pseudogap minimizing the total exchange and kinetic energy Charge-spin entanglement induced by phase string AFM state - +

Summary Pseudogap state is firmly established by experiment as one of the most exotic phases in the cuprates which is closely related to high-T c superconductivity Doped Mott insulator/antiferromagnet provides a suitable microscopic model to understand the pseudogap physics Mott constraint leads to a new sign structure greatly reduced from the fermion signs at low doping