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High Temperature Superconductivity: Outline: S. Kivelson V. Emery E. Carlson M. Granath V. Oganesyan X-J. Zhou Z-X. Shen Basic facts concerning the cuprates Stripes: What are they and why do they occur Experimental signatures of stripes Are stripes good or bad for superconductivity ? Consequences of stripe formation: • Fractionalization • Confinement D. Orgad Racah Institute, Hebrew University, Jerusalem The Cuprates: Basic Structure La(2 - x)SrxCuO4 • Universal element – CuO planes • Parent (undoped) compounds – Heisenberg antiferromagnets • Hole doping by chemical substitution / Oxygen doping The Cuprates: Typical Phase Diagram Renner et al. Harris et al. Warren et al. Takagi et al. ARPES NMR DC resistivity UD Bi2212 T tunneling Puchkov et al. AF Pseudogap SC under optimal doping over x Optical conductivity Neutron scattering, Specific heat … The Central Question: What happens to an AF upon doping with holes? Holes in an AF : Why Do Stripes Occur? t J x t Coulomb Interactions PHASE SEPARATION Kinetic Energy H t Frustration ( cis c js ij s STRIPES ni n j h.c.) J ( Si S j ) 4 ij Stripes in Other Systems: Competing Interactions Ferrofluid between glass plates Ferromagnetic garnet film l~1cm l~10mm l~10mm l~400A Ferromagnetic garnet film Block copolymers film Stripe Signatures in S(k,w) Real Space Momentum Space ky kx lss l ls l lccc 2 ls Experimental Evidence for Stripes: Neutron Scattering k y Static stripe order (LNSCO) kx 0.25 E=24.5meV Dynamic stripes (YBCO) Mook et al. Tranquada et al. 0.12 Experimental Evidence for Stripes: ARPES Angle Resolved PhotoEmission Spectroscopy measures the single hole spectral function A (k ,w ) dx dt ei ( kxw t ) ( x, t )(0,0) n(k ) dw A (k ,w ) LNSCO Experimental Evidence for Stripes: Tunneling Microscopy B=5T B=0 Howald et al. Hoffman et al. Consequences of Stripe Formation: Spin-gap and Enhanced SC Correlations Doped Spin Ladders: known to be spin-gapped s Je w L R RL T cos( 2 s )e i 2c s e i 2 c AF Stripes PG SC x • The spin-gap creates an amplitude of the SC order parameter • Provides high pairing scale (avoid Coulomb repulsion) A Problem … Good News: In 1D a spin-gap enhances pairing: divergent for Kc>1/2 (Kc<1 for repulsive interactions) 1 /( 2 K C 1 ) T c ~ E F ( g SC / E F ) Bad News: It also enhances CDW correlations: more divergent ! sc sT ( 2 Kc1 ) CDW sT ( 2 K c ) 1 /( 2 K C ) T c ~ E F ( g CDW / E F ) Old problem from search for organic superconductors … And Its Resolution T Stripe fluctuations (quantum, thermal or quenched) are necessary for high Tc! y2 y1 Nematic? L L2 1 Phase Phase Stiffness PG Stiffness AF x SC y static fluctuating x dissolved Stripe fluctuations dephase CDW coupling: e Yamada et al. 2ik F ( L1 L2 ) Stripe fluctuations enhance phase coupling: e | y1 y2 | e 2 k F L2 2 y 2 e2 Consequences of Stripe Formation: Electron Fractionalization Above Tc In a Fermi liquid the elementary w excitations have the quantum numbers of an electron Mo surface k w v F k state multi-qp background Valla et al. qp peak In a Luttinger liquid the excitations come in 4 flavors RL cs w EDC k MDC MDC (w 0) EDC ( k 0) w vc | k | c 0 c 0.3 w w vs | k | c 0.5 ( L, c ) w v sk w v ck ( L, s ) | w | v c k | w | v s k k Evidence for Fractionalization ARPES in La1.25Nd0.6Sr0.15CuO4 Breakdown of W-F Law 2 kB L0 T 3 e 2 1DEG s 0 , c 0.5 v s 0.7 eV A in Pr1.85Ce0.15CuO4 v c 3.5 eV A Orgad et al. Sharp in Momentum Broad in Energy Hill et al. Below Tc: A Coherent Peak Optimally Doped BSCCO (Tc=91K) Not a Conventional QP • Not present above Tc • Intensity grows below Tc • Energy and lifetime not temperature dependent Fedorov et al. Josephson Coupling Confines 1D Solitons The electronic operator L e s and c i 2 c c s s ) s , c creates kinks in 2 x Charge and spin solitons create phase shift in pair field cos( 2 s )e i 2 c s c Frustrated Josephson Coupling H ijJosephson J SC [ i j h.c.] between solitons Bound spin-charge soliton pair < A (k,w) in the Superconducting Phase A (k ,w ) Z (w E ) incoherent • Quasiparticle weight depends on superfluid density: Z (T , x ) (T , x ) ( 2 c 1 / 2 ) Feng et al. Conclusions • Stripes are ubiquitous in the cuprate high temperature superconductors • They are important for high temperature superconductivity • There is evidence that the normal state of the cuprates is fractionalized • In a quasi-one-dimensional superconductor Tc also marks a confinement transition Landau Theory of Stripe Phases Coupled charge (CDW) order k and spin (SDW) order SQ q , a a * 2 1 2 4 1 2 4 F r | k | U | k | rS | SQ q | U S | SQ q | U x | SQ q SQ q | 2 2 l1[( SQ q SQ q ) k h.c.] l2 | SQ q |2 | k |2 k 2q Stripes are “charge driven” : 0 S 0 Spin order is secondary and may be absent Zachar et al. Spin-gap Proximity Effect ~ kF kF Single particle tunneling irrelevant “system” “environment” Pair tunneling K K F When pair F K~ K~ F 1 ~ ~ ~ Ks Ks 1 4 Kc Kc F possible tunneling kF ~ kF ~ ~ H pair t cos( 2 s ) cos( 2 s ) cos[ 2 ( c c )] is relevant. The spin modes and the relative charge phase mode are gapped. The only gapless mode involves the total SC phase c ~c • Kinetic energy driven pairing • Repulsive interactions within system and environment increase • Repulsive interactions between system and environment decrease • Pre-existing spin-gap in environment decreases ARPES and Stripes Angle Resolved PhotoEmission Spectroscopy measures the single hole spectral function A (k ,w ) dx dt ei ( kxw t ) ( x, t )(0,0) n(k ) dw A ( k , w ) LNSCO LNSCO LSCO m dw A ( k , w ) m 30 meV Zhou et al. Disordered Stripe Array: Spectral Weight Low Energy Spectral Weight ( ) m 1 dw eik ( r r ') n ( r )n ( r ' ) (w En ) S r ,r ' n m 0.2 ( ) Granath et al. ( ) Disordered Stripe Array: Interacting Spectral Function Granath et al. A Model: Quasi-one-dimensional Superconductor Charge: Gapless Spin: Gapped Weak Pair Tunneling (Couples charge and spin) Prediction: New Magnetic Resonance Neutron scattering measures the spin-spin correlation function: dx dt e i ( k x w t ) S2 k F ( x, t ) S2k (0,0) F 1 S2k R ' L ' creates 2 spin solitons and 2 charge solitons F 2 , ' Treat more massive spin solitons as static and solve for the charges: s s 2 c (x ) Hc vc [K c ( x c )2 ( xc ) 2 Kc ] c ( x) cos( 2 c ) Get effective Schrodinger equation for spins: H eff vs2 2 2 2 s V ( x1 x2 ) 2 2 s j 1 x j •Spin 1 mode that exists below 0.4 Tc ,0 •2kF mode: should appear around 2 •Threshold at 2s