High Temperature Superconductivity: :Outline S. Kivelson V. Emery
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 T .Harris et al .Warren et al ARPES NMR .Takagi et al UD Bi2212 tunneling DC .resistivity 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 PHASE SEPARATION Coulomb Interactions STRIPES Kinetic Energy ni n j Frustration H t ij s( cis c js h.c.) J ij ( Si S j 4 )
Stripes in Other Systems: Competing Interactions Ferrofluid between glass plates Ferromagnetic garnet film cm m m Ferromagnetic garnet film Block copolymers film Stripe Signatures in S(k, Real Space Momentum Space ky kx
ccc ss s 2 s Experimental Evidence for Stripes: Neutron Scattering k y kx Static stripe order (LNSCO) 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 functionA ( k , ) dx dt ei ( kx t ) ( x, t )(0,0) n ( k ) d A ( k , ) LNSCO Experimental Evidence for Stripes: Tunneling Microscopy B=0 B=5T .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 L R R L w T cos( 2 s )e i 2 c se 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( sc sT ( 2 K c 1 ) 1 /( 2 K C 1 ) T c ~ E F ( g SC / E F ) :Bad News :It also enhances CDW correlations more divergent ! CDW
sT ( 2 Kc ) 1 /( 2 K g ~ ( / ) T c E F CDW E F C ) 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 L1 L2 Phase StiffnesPhase Stiffness s PG 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 e 2 k F L2 2 y 2 2 Consequences of Stripe Formation: Electron Fractionalization Above Tc In a Fermi liquid the elementary excitations have the quantum numbers of an electron Mo surface k state v F k
multi-qp background .Valla et al qp peak k MDC MDC ( 0) EDC ( k 0) v c | k | c 0 c 0.3 In a Luttinger liquid the excitations come in flavors 4 RL cs EDC v s | k | c 0.5
( L, c ) v s k v c k ( L, s ) | |v c k | |v s k k Evidence for Fractionalization ARPES in La1.25Nd0.6Sr0.15CuO4 Breakdown of W-F Law 2 1DEG s 0 , c 0.5 v s 0.7 eV A 2 kB L0 T 3 e 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
creates kinks in s , c 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 in the Superconducting Phase A ( k , ) Z ( E ) incoherent Quasiparticle weight depends on superfluid density: Z (T , x ) (T , x ) ( 2c 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 SQq , 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 SQq | 2
2 1[( SQq SQq ) k h.c.] 2 | SQq |2 | k |2 k 2q Stripes are charge driven : 0 S 0 Spin order is secondary and may be absent .Zachar et al ~ k F k F Spin-gap Proximity Effect 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 functionA ( k , ) dx dt ei ( kx t ) ( x, t )(0,0) LNSCO n(k ) d A ( k , ) LNSCO LSCO
d A ( k , ) 30 meV .Zhou et al Disordered Stripe Array: Spectral Weight Low Energy Spectral Weight ( ) 1 d eik ( r r ') n ( r ) n ( r ' ) ( En ) S r ,r ' n 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 t )
S2k F ( x, t ) S 2 k (0,0) F 1 S 2 k 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 c (x ) v ( ) 2 Hc c
2 [K c ( x c )2 x c Kc ] c ( x ) cos( 2 c ) Get effective Schrodinger equation for spins: v s 2 2 2 eff H 2 s 2 s j 1 x j 2
V ( x1 x2 ) Spin 1 mode that exists below 0.4 Tc 2kF mode: should appear around Threshold at 2s ,0 2
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