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**Concepts in High Temperature Superconductivity**

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500 references V. J. Emery, S. A. Kivelson, E.Arrigoni, I.Bindloss, E.Carlson, S.Chakravarty, L.Chayes, K.Fabricius, E.Fradkin, M.Granath, D-H.Lee, H-Q. Lin, U.Low, T.Lubensky, E.Manousakis, Z.Nussinov, V.Oganesyan, D.Orgad, L.Pryadko, M.Salkola, Z-X.Shen, J.Tranquada, O.Zachar

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**Superconductivity Fermions: Pauli exclusion principle**

Bosons can condense Stable phase of matter Macroscopic Quantum Behavior Pair wavefunction acts like “order parameter”

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**Superconductivity Fermions: Pauli exclusion principle**

Bosons can condense Stable phase of matter Macroscopic Quantum Behavior Pair wavefunction acts like “order parameter”

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**Conventional Superconductivity**

John Bardeen Leon Cooper Bob Schrieffer Conventional Superconductivity BCS Theory Instability of the metallic state

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**Simple Metals: The Fermi Gas**

Fermi Surface Free Electrons E ~ k2 Pauli exclusion principle Fill to Fermi level Adiabatically turn on temperature, most states unaffected Very dilute gas of excitations: quasiparticles KE most important Pauli exclusion principle fill to Fermi level Fermi sea Stable except for …

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**Simple Metals: The Fermi Liquid**

Fermi Surface Pauli exclusion principle Fill to Fermi level Adiabatically turn on temperature, most states unaffected Very dilute gas of excitations: quasiparticles Fermi Gas + Interactions Fermi Liquid Kinetic Energy Dominant Quasiparticles = Dressed Electrons

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**Retardation is Essential**

Cooper Pairing Fermi Liquid Unstable to Pairing Pair electrons near Fermi Surface Phonon mediated Retardation is Essential to overcome Coulomb repulsion

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**Of A Tranquil Fermi Sea—**

BCS Haiku: Instability Of A Tranquil Fermi Sea— Broken Symmetry

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**High Temperature Superconductors**

HgCuO YBCO7 LSCO

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**High Temperature Superconductors**

Copper Oxygen Planes Other Layers Layered structure quasi-2D system Layered structure quasi-2D system

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**High Temperature Superconductors**

Copper-Oxygen Planes Important “Undoped” is half-filled Antiferromagnet Naive band theory fails Strongly correlated Oxygen Copper

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**High Temperature Superconductors**

Dope with holes Superconducts at certain dopings T AF SC x Oxygen Copper

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**Mysteries of High Temperature Superconductivity**

Ceramic! (Brittle) “Non-Fermi liquid” normal state Magnetism nearby (antiferromagnetism) Make your own (robust) Pseudogap Phase ordering transition

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**Two Energy Scales in a Superconductor**

Two component order parameter Amplitude Pairing Gap Phase Phase Coherence Superfluid Density

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**BCS is a mean field theory in which pairing precipitates order**

42 38 Y-123 (ud) 62 104 Bi-2212 (od) 140 90 116 Y-123 (op) 55 Y-123 (od) 60 95 220 Bi-2212 (op) 83 275 Bi-2212 (ud) 25 26 Tl-2201 (od) 160 80 91 122 Tl-2201 (op) 130 133 435 Hg-1223 (op) 108 290 Hg-1212 (op) 180 96 192 Hg-1201 (op) 100 20 LSCO (od) 54 58 LSCO (op) 47 30 75 LSCO (ud) Material Phase Fluctuations Important in Cuprates Emery, Kivelson, Nature, 374, 434 (1995) EC, Kivelson, Emery, Manousakis, PRL 83, 612 (1999) Material Pb 7.9 7.2 6X105 Nb3Sn 18.7 17.8 2X104 UBe13 0.8 0.9 102 BaKBiO 17.4 26 5X102 K3C60 26 20 102 MgB2 15 39 1.4X103

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**Tc and the Energy Scales**

superconductivity AF x BCS: ~ 1000 Tc HTSC: ~ Tc underdoped

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**Mysteries of High Temperature Superconductivity**

Ceramic! (Brittle) “Non-Fermi liquid” normal state Magnetism nearby (antiferromagnetism) Make your own (robust) Pseudogap Phase ordering transition BCS to the rescue?

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**There is no room for retardation in the cuprates**

BCS: Cuprates: How do we get a high pairing scale despite the strong Coulomb repulsion?

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**A Fermi Surface Instability Requires a Fermi Surface!**

How do we get superconductivity from a non-Fermi liquid?

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**Strong Correlation Fermi Liquid k-space structure**

Kinetic energy is minimized Pairing is potential energy driven Real space structure Kinetic energy is highly frustrated System works to relieve KE frustration

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**Doped Antiferromagnets**

Hole Motion is Frustrated

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**Doped Antiferromagnets**

Compromise # 1: Phase Separation Relieves some KE frustration Pure AF Hole Rich Like Salt Crystallizing From Salt Water, The Precipitate (AF) is Pure

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**Coulomb Frustrated Phase Separation**

Long range homogeneity Short range phase separation Compromise # 2: mesoscale structure Patches interleave quasi-1D structure – stripes ? Hole Rich Poor

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**Stripes Rivers of charge between antiferromagnetic strips**

Electronic structure becomes effectively 1D

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**Competition often produces stripes**

Ferromagnetic garnet film Faraday effect Period ~ m Ferrofluid confined between two glass plates Period ~ 1cm Ferromagnetic garnet film Period ~ m Block copolymers Period ~ 4X10-8 m

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**What’s so special about 1D?**

canal John Scott Russell's Soliton Wave Re-created On Wednesday 12 July 1995, an international gathering of scientists witnessed a re-creation of the famous 1834 'first' sighting of a soliton or solitary wave on the Union Canal near Edinburgh. They were attending a conference on nonlinear waves in physics and biology at Heriot-Watt University, near the canal. The occasion was part of a ceremony to name a new aqueduct after John Scott Russell, the Scottish scientist who made the original observation. The aqueduct carries the Union Canal over the Edinburgh City Bypass. In 1834, John Scott Russell was observing a boat being drawn along 'rapidly' by a pair of horses. When the boat suddenly stopped Scott Russell noticed that the bow wave continued forward "at great velocity, assuming the form of a large solitary elevation, a well-defined heap of water which continued its course along the channel apparently without change of form or diminution of speed". Intrigued, the young scientist followed the wave on horseback as it rolled on at about eight or nine miles an hour, but after a chase of one or two miles he lost it. What’s so special about 1D?

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**1D: Spin-Charge Separation**

spin excitation charge excitation point out topological, 1D special, point-like, and higher D different.

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**1D: “Bosonization transformation”**

Expresses fermions as kinks in boson fields: spin soliton electron = + charge soliton Hamiltonian of interacting fermions becomes Hamiltonian of non-interacting bosons Spin Charge Separation Electron No Longer Exists! Non-Fermi Liquid (Luttinger Liquid)

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**Advantages of a quasi-1D Superconductor**

non-Fermi liquid strongly correlated controlled calculations

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**Kinetic Energy Driven Pairing?**

Proximity Effect superconductor metal Individually, free energies minimized Metal pairs (at a cost!) to minimize kinetic energy across the barrier

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**Spin Charge Separation**

1D spin-charge separation Pair spins only Avoid Coulomb Repulsion!

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**Spin Gap Proximity Effect**

Kinetic energy driven pairing in a quasi-1D superconductor Metallic charge stripe acquires spin gap through communication with gapped environment

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**Spin Gap Proximity Effect**

Kinetic energy driven pairing in a quasi-1D superconductor kF = k’F Conserve momentum and energy Single particle tunneling is irrelevant kF k’F

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**Spin Gap Proximity Effect**

Kinetic energy driven pairing in a quasi-1D superconductor kF = k’F Conserve momentum and energy Single particle tunneling is irrelevant But pairs of zero total momentum can tunnel at low energy pairs form to reduce kinetic energy Step 1: Pairing kF k’F

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**1D is Special Spin Gap = CDW Gap = Superconducting Gap Which will win?**

CDW stronger for repulsive interactions (Kc<1) Don’t Make a High Temperature Insulator!

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**Stripe Fluctuations Step 2: Phase Coherence Frustrated Favorable**

Discourage CDW Stripe fluctuations Encourage SC

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**Inherent Competition think local stripe order**

superconductivity T Josephson coupling Pairing think local stripe order think “stripe phonon” stripe fluctuations Static Stripes Good pairing Bad phase coherence Fluctuating Stripes Bad pairing Good phase coherence

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**Behavior of a Quasi-1D Superconductor**

Treat rivers of charge as 1D objects

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**Behavior of a Quasi-1D Superconductor**

High Temperature Effective Dimension = 1 Spin charge separation on the Rivers of Charge Electron dissolves Non-Fermi Liquid (Luttinger Liquid) Intermediate Temperature Effective Dimension = 1 Rivers of charge acquire spin gap from local environment Pseudogap Non-Fermi Liquid (Luther-Emery Liquid) Low Temperature Effective Dimension = 3 1D system cannot order Dimensional Crossover Pair tunneling between rivers produces phase coherence Electron recombines

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**Dimensional Crossover and the Quasiparticle**

Chains coupled in superconducting state c o n f i n e m e n t s c e Superconducting order parameter switches sign across each soliton

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**Dimensional Crossover and the Quasiparticle**

c o n f i n e m e n t s c Bound state of spin and charge Electron is stable in superconducting state

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**Crossover Diagram of a Quasi-1D Superconductor**

Do/2 1D Luttinger Liquid 1D Luther-Emery Liquid 3D Fermi Liquid T Pseudogap 3D Superconductor Gap t (increases with x?) Crossover Diagram of a Quasi-1D Superconductor

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**Is mesoscale structure necessary for high Tc superconductivity?**

Mesoscale structure can enhance pairing (spin-gap) May be necessary to get pairing from repulsion Always bad for phase ordering

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**Conclusions Strongly correlated Kinetic Energy driven order**

Formation of mesoscale structure Formation of pairs by proximity effect Phase coherence by pair tunneling between stripes Quasi-1D Superconductor (Controlled calculations) Non-Fermi liquid normal state Pseudogap state Strong pairing from repulsive interactions Inherent competition between stripes and superconductivity Dimensional crossover to superconducting state Strong correlation leads to all sorts of new physics. I focused on 3 that can occur. “Controlled” is anthropomorphic

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