the role of charge order in the mechanism of high...
TRANSCRIPT
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The Role of Charge Order in the Mechanism of
High Temperature SuperconductivityTalk at the Oak Ridge National Laboratory, March 28, 2008
Eduardo Fradkin
Department of PhysicsUniversity of Illinois at Urbana Champaign
March 28, 2008
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Collaborators
◮ Steven Kivelson (Stanford University) and Victor J. Emery∗
(Brookhaven National Laboratory)
◮ Enrico Arrigoni (TU Graz, Austria), Vadim Oganesyan (YaleUniversity), Erez Berg (Stanford), Eun-Ah Kim (Stanford),Shoucheng Zhang (Stanford), Michael Lawler (Toronto), DanielBarci (UERJ, Brazil), Tom Lubensky (U. Penn),Erica Carlson(Purdue), Karin Dahmen (UIUC), Kai Sun (UIUC), E. Manousakis(FSU/NHMFL)
◮ John Tranquada (Brookhaven), Aharon Kapitulnik (Stanford), PeterAbbamonte (UIUC), Lance Cooper (UIUC)
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References
◮ Steven A. Kivelson and Eduardo Fradkin, How optimalinhomogeneity produces high temperature superconductivity;chapter 15 of Treatise of High Temperature Superconductivity, J.Robert Schrieffer and J. Brooks, editors, Springer-Verlag (2007);arXiv:cond-mat/0507459.
◮ Steven A. Kivelson, Eduardo Fradkin, Vadim Oganesyan, IanBindloss, John Tranquada, Aharon Kapitulnik and Craig Howald,How to detect fluctuating stripes in high temperaturesuperconductors, Rev. Mod. Phys. 75, 1201 (2003);arXiv:cond-mat/02010683.
◮ Steven A. Kivelson, Eduardo Fradkin and Victor J. Emery, ElectronicLiquid Crystal Phases of a Doped Mott Insulator, Nature 393, 550(1998); arXiv:cond-mat/9707327.
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Outline
◮ Electronic liquid crystal phases are unavoidable in doped Mottinsulators
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Outline
◮ Electronic liquid crystal phases are unavoidable in doped Mottinsulators
◮ Experimental evidence for electronic liquid crystal phases in the highTc superconductors, Sr3Ru2O7, and quantum Hall systems
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Outline
◮ Electronic liquid crystal phases are unavoidable in doped Mottinsulators
◮ Experimental evidence for electronic liquid crystal phases in the highTc superconductors, Sr3Ru2O7, and quantum Hall systems
◮ Charge Order and Superconductivity: friend or foe?
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Outline
◮ Electronic liquid crystal phases are unavoidable in doped Mottinsulators
◮ Experimental evidence for electronic liquid crystal phases in the highTc superconductors, Sr3Ru2O7, and quantum Hall systems
◮ Charge Order and Superconductivity: friend or foe?
◮ If inhomogeneous phases are part of the mechanism of hightemperature superconductivity, is there an optimal degree ofinhomogeneity?
![Page 8: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/8.jpg)
Outline
◮ Electronic liquid crystal phases are unavoidable in doped Mottinsulators
◮ Experimental evidence for electronic liquid crystal phases in the highTc superconductors, Sr3Ru2O7, and quantum Hall systems
◮ Charge Order and Superconductivity: friend or foe?
◮ If inhomogeneous phases are part of the mechanism of hightemperature superconductivity, is there an optimal degree ofinhomogeneity?
◮ High Tc Superconductivity in a Striped Hubbard Model
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Outline
◮ Electronic liquid crystal phases are unavoidable in doped Mottinsulators
◮ Experimental evidence for electronic liquid crystal phases in the highTc superconductors, Sr3Ru2O7, and quantum Hall systems
◮ Charge Order and Superconductivity: friend or foe?
◮ If inhomogeneous phases are part of the mechanism of hightemperature superconductivity, is there an optimal degree ofinhomogeneity?
◮ High Tc Superconductivity in a Striped Hubbard Model
◮ Isolated doped Hubbard Ladders as the prototype spin-gap systems
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Outline
◮ Electronic liquid crystal phases are unavoidable in doped Mottinsulators
◮ Experimental evidence for electronic liquid crystal phases in the highTc superconductors, Sr3Ru2O7, and quantum Hall systems
◮ Charge Order and Superconductivity: friend or foe?
◮ If inhomogeneous phases are part of the mechanism of hightemperature superconductivity, is there an optimal degree ofinhomogeneity?
◮ High Tc Superconductivity in a Striped Hubbard Model
◮ Isolated doped Hubbard Ladders as the prototype spin-gap systems
◮ Estimating Tc : How high is high?
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Outline
◮ Electronic liquid crystal phases are unavoidable in doped Mottinsulators
◮ Experimental evidence for electronic liquid crystal phases in the highTc superconductors, Sr3Ru2O7, and quantum Hall systems
◮ Charge Order and Superconductivity: friend or foe?
◮ If inhomogeneous phases are part of the mechanism of hightemperature superconductivity, is there an optimal degree ofinhomogeneity?
◮ High Tc Superconductivity in a Striped Hubbard Model
◮ Isolated doped Hubbard Ladders as the prototype spin-gap systems
◮ Estimating Tc : How high is high?
◮ Conclusions and Open Questions
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Outline
◮ Electronic liquid crystal phases are unavoidable in doped Mottinsulators
◮ Experimental evidence for electronic liquid crystal phases in the highTc superconductors, Sr3Ru2O7, and quantum Hall systems
◮ Charge Order and Superconductivity: friend or foe?
◮ If inhomogeneous phases are part of the mechanism of hightemperature superconductivity, is there an optimal degree ofinhomogeneity?
◮ High Tc Superconductivity in a Striped Hubbard Model
◮ Isolated doped Hubbard Ladders as the prototype spin-gap systems
◮ Estimating Tc : How high is high?
◮ Conclusions and Open Questions
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Phase Diagram of the High Tc Superconductors
T
x
Mott
Insu
lato
r
“pseudogap”
Superconductor
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Electron Liquid Crystal Phases
Steven A. Kivelson, Eduardo Fradkin and Victor J. EmeryElectronic Liquid Crystal Phases of a Doped Mott InsulatorNature 393, 550 (1998); arXiv:cond-mat/9707327
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Electron Liquid Crystal Phases
Steven A. Kivelson, Eduardo Fradkin and Victor J. EmeryElectronic Liquid Crystal Phases of a Doped Mott InsulatorNature 393, 550 (1998); arXiv:cond-mat/9707327
◮ Doping a Mott insulator: inhomogeneous phases arise due to thecompetition between phase separation and strong correlations
![Page 16: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/16.jpg)
Electron Liquid Crystal Phases
Steven A. Kivelson, Eduardo Fradkin and Victor J. EmeryElectronic Liquid Crystal Phases of a Doped Mott InsulatorNature 393, 550 (1998); arXiv:cond-mat/9707327
◮ Doping a Mott insulator: inhomogeneous phases arise due to thecompetition between phase separation and strong correlations
◮ Crystal Phases: break all continuous translation symmetries androtations
![Page 17: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/17.jpg)
Electron Liquid Crystal Phases
Steven A. Kivelson, Eduardo Fradkin and Victor J. EmeryElectronic Liquid Crystal Phases of a Doped Mott InsulatorNature 393, 550 (1998); arXiv:cond-mat/9707327
◮ Doping a Mott insulator: inhomogeneous phases arise due to thecompetition between phase separation and strong correlations
◮ Crystal Phases: break all continuous translation symmetries androtations
◮ Smectic (Stripe) phases: break one translation symmetry androtations
![Page 18: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/18.jpg)
Electron Liquid Crystal Phases
Steven A. Kivelson, Eduardo Fradkin and Victor J. EmeryElectronic Liquid Crystal Phases of a Doped Mott InsulatorNature 393, 550 (1998); arXiv:cond-mat/9707327
◮ Doping a Mott insulator: inhomogeneous phases arise due to thecompetition between phase separation and strong correlations
◮ Crystal Phases: break all continuous translation symmetries androtations
◮ Smectic (Stripe) phases: break one translation symmetry androtations
◮ Nematic and Hexatic Phases: are uniform and anisotropic
![Page 19: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/19.jpg)
Electron Liquid Crystal Phases
Steven A. Kivelson, Eduardo Fradkin and Victor J. EmeryElectronic Liquid Crystal Phases of a Doped Mott InsulatorNature 393, 550 (1998); arXiv:cond-mat/9707327
◮ Doping a Mott insulator: inhomogeneous phases arise due to thecompetition between phase separation and strong correlations
◮ Crystal Phases: break all continuous translation symmetries androtations
◮ Smectic (Stripe) phases: break one translation symmetry androtations
◮ Nematic and Hexatic Phases: are uniform and anisotropic◮ Uniform fluids: break no spatial symmetries
![Page 20: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/20.jpg)
Electron Liquid Crystal Phases
Steven A. Kivelson, Eduardo Fradkin and Victor J. EmeryElectronic Liquid Crystal Phases of a Doped Mott InsulatorNature 393, 550 (1998); arXiv:cond-mat/9707327
◮ Doping a Mott insulator: inhomogeneous phases arise due to thecompetition between phase separation and strong correlations
◮ Crystal Phases: break all continuous translation symmetries androtations
◮ Smectic (Stripe) phases: break one translation symmetry androtations
◮ Nematic and Hexatic Phases: are uniform and anisotropic◮ Uniform fluids: break no spatial symmetries
◮ ◮ High Tc Superconductors : Lattice effects ⇒ breaking of pointgroup symmetries
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Electron Liquid Crystal Phases
Steven A. Kivelson, Eduardo Fradkin and Victor J. EmeryElectronic Liquid Crystal Phases of a Doped Mott InsulatorNature 393, 550 (1998); arXiv:cond-mat/9707327
◮ Doping a Mott insulator: inhomogeneous phases arise due to thecompetition between phase separation and strong correlations
◮ Crystal Phases: break all continuous translation symmetries androtations
◮ Smectic (Stripe) phases: break one translation symmetry androtations
◮ Nematic and Hexatic Phases: are uniform and anisotropic◮ Uniform fluids: break no spatial symmetries
◮ ◮ High Tc Superconductors : Lattice effects ⇒ breaking of pointgroup symmetries
◮ If lattice effects are weak (high T ) ⇒ continuous symmetriesessentially recovered
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Electron Liquid Crystal Phases
Steven A. Kivelson, Eduardo Fradkin and Victor J. EmeryElectronic Liquid Crystal Phases of a Doped Mott InsulatorNature 393, 550 (1998); arXiv:cond-mat/9707327
◮ Doping a Mott insulator: inhomogeneous phases arise due to thecompetition between phase separation and strong correlations
◮ Crystal Phases: break all continuous translation symmetries androtations
◮ Smectic (Stripe) phases: break one translation symmetry androtations
◮ Nematic and Hexatic Phases: are uniform and anisotropic◮ Uniform fluids: break no spatial symmetries
◮ ◮ High Tc Superconductors : Lattice effects ⇒ breaking of pointgroup symmetries
◮ If lattice effects are weak (high T ) ⇒ continuous symmetriesessentially recovered
◮ 2DEG in GaAs heterostructures ⇒ continuous symmetries
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Electron Liquid Crystal Phases
Steven A. Kivelson, Eduardo Fradkin and Victor J. EmeryElectronic Liquid Crystal Phases of a Doped Mott InsulatorNature 393, 550 (1998); arXiv:cond-mat/9707327
◮ Doping a Mott insulator: inhomogeneous phases arise due to thecompetition between phase separation and strong correlations
◮ Crystal Phases: break all continuous translation symmetries androtations
◮ Smectic (Stripe) phases: break one translation symmetry androtations
◮ Nematic and Hexatic Phases: are uniform and anisotropic◮ Uniform fluids: break no spatial symmetries
◮ ◮ High Tc Superconductors : Lattice effects ⇒ breaking of pointgroup symmetries
◮ If lattice effects are weak (high T ) ⇒ continuous symmetriesessentially recovered
◮ 2DEG in GaAs heterostructures ⇒ continuous symmetries
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Electronic Liquid Crystal Phases in High Tc
Superconductors
◮ Liquid: Isotropic, breaks no spacial symmetries; either a conductoror a superconductor
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Electronic Liquid Crystal Phases in High Tc
Superconductors
◮ Liquid: Isotropic, breaks no spacial symmetries; either a conductoror a superconductor
◮ Nematic: Lattice effects reduce the symmetry to a rotations by π
2
(“Ising”); translation and reflection symmetries are unbroken; it is ananisotropic liquid with a preferred axis
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Electronic Liquid Crystal Phases in High Tc
Superconductors
◮ Liquid: Isotropic, breaks no spacial symmetries; either a conductoror a superconductor
◮ Nematic: Lattice effects reduce the symmetry to a rotations by π
2
(“Ising”); translation and reflection symmetries are unbroken; it is ananisotropic liquid with a preferred axis
◮ Smectic: breaks translation symmetry only in one direction butliquid-like on the other; Stripe phase; (infinite) anisotropy ofconductivity tensor
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Electronic Liquid Crystal Phases in High Tc
Superconductors
◮ Liquid: Isotropic, breaks no spacial symmetries; either a conductoror a superconductor
◮ Nematic: Lattice effects reduce the symmetry to a rotations by π
2
(“Ising”); translation and reflection symmetries are unbroken; it is ananisotropic liquid with a preferred axis
◮ Smectic: breaks translation symmetry only in one direction butliquid-like on the other; Stripe phase; (infinite) anisotropy ofconductivity tensor
◮ Crystal(s): electron solids (“CDW”); insulating states
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Electronic Liquid Crystal Phases in High Tc
Superconductors
◮ Liquid: Isotropic, breaks no spacial symmetries; either a conductoror a superconductor
◮ Nematic: Lattice effects reduce the symmetry to a rotations by π
2
(“Ising”); translation and reflection symmetries are unbroken; it is ananisotropic liquid with a preferred axis
◮ Smectic: breaks translation symmetry only in one direction butliquid-like on the other; Stripe phase; (infinite) anisotropy ofconductivity tensor
◮ Crystal(s): electron solids (“CDW”); insulating states
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Electronic Liquid Crystal Phases in HTSC, cont’d
Soft Quantum Condensed Matter!
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Schematic Phase Diagram of Doped Mott Insulators
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Order Parameters for the Smectic (Stripe) State
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Order Parameters for the Smectic (Stripe) State
◮ Two-dimensionally ordered state with unidirectional CDW order
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Order Parameters for the Smectic (Stripe) State
◮ Two-dimensionally ordered state with unidirectional CDW order
◮ charge modulation ⇒ charge stripe
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Order Parameters for the Smectic (Stripe) State
◮ Two-dimensionally ordered state with unidirectional CDW order
◮ charge modulation ⇒ charge stripe
◮ if it coexists with spin order ⇒ spin stripe
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Order Parameters for the Smectic (Stripe) State
◮ Two-dimensionally ordered state with unidirectional CDW order
◮ charge modulation ⇒ charge stripe
◮ if it coexists with spin order ⇒ spin stripe
◮ stripe state ⇒ new Bragg peaks of the electron density at
~k = ±~Qch = ± 2π
λch
ex
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Order Parameters for the Smectic (Stripe) State
◮ Two-dimensionally ordered state with unidirectional CDW order
◮ charge modulation ⇒ charge stripe
◮ if it coexists with spin order ⇒ spin stripe
◮ stripe state ⇒ new Bragg peaks of the electron density at
~k = ±~Qch = ± 2π
λch
ex
◮ spin stripe ⇒ magnetic Bragg peaks at
~k = ~Qs = (π, π) ± 1
2~Qch
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Order Parameters for the Smectic (Stripe) State
◮ Two-dimensionally ordered state with unidirectional CDW order
◮ charge modulation ⇒ charge stripe
◮ if it coexists with spin order ⇒ spin stripe
◮ stripe state ⇒ new Bragg peaks of the electron density at
~k = ±~Qch = ± 2π
λch
ex
◮ spin stripe ⇒ magnetic Bragg peaks at
~k = ~Qs = (π, π) ± 1
2~Qch
◮ Order Parameter: 〈n~Qch
〉, Fourier component of the electron density
at ~Qch
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Order Parameters for the Smectic (Stripe) State
◮ Two-dimensionally ordered state with unidirectional CDW order
◮ charge modulation ⇒ charge stripe
◮ if it coexists with spin order ⇒ spin stripe
◮ stripe state ⇒ new Bragg peaks of the electron density at
~k = ±~Qch = ± 2π
λch
ex
◮ spin stripe ⇒ magnetic Bragg peaks at
~k = ~Qs = (π, π) ± 1
2~Qch
◮ Order Parameter: 〈n~Qch
〉, Fourier component of the electron density
at ~Qch
◮ Smectic (stripe) order couples linearly to disorder ⇒ macroscopicglassy behavior
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Order Parameters for the Smectic (Stripe) State
◮ Two-dimensionally ordered state with unidirectional CDW order
◮ charge modulation ⇒ charge stripe
◮ if it coexists with spin order ⇒ spin stripe
◮ stripe state ⇒ new Bragg peaks of the electron density at
~k = ±~Qch = ± 2π
λch
ex
◮ spin stripe ⇒ magnetic Bragg peaks at
~k = ~Qs = (π, π) ± 1
2~Qch
◮ Order Parameter: 〈n~Qch
〉, Fourier component of the electron density
at ~Qch
◮ Smectic (stripe) order couples linearly to disorder ⇒ macroscopicglassy behavior
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Order Parameters for the Nematic State
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Order Parameters for the Nematic State◮ Two-dimensionally translationally invariant metallic state with
spontaneously broken rotational symmetry, i.e. spontaneous“orthorhombicity”
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Order Parameters for the Nematic State◮ Two-dimensionally translationally invariant metallic state with
spontaneously broken rotational symmetry, i.e. spontaneous“orthorhombicity”
◮ Weak coupling picture: spontaneous quadrupolar distortion of theFermi surface; a Pomeranchuk instabilityVadim Oganesyan, Steven A. Kivelson, and Eduardo Fradkin,Quantum Theory of a Nematic Fermi Fluid, Phys. Rev. B 64,195109 (2001); arXiv:cond-mat/0102093.
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Order Parameters for the Nematic State◮ Two-dimensionally translationally invariant metallic state with
spontaneously broken rotational symmetry, i.e. spontaneous“orthorhombicity”
◮ Weak coupling picture: spontaneous quadrupolar distortion of theFermi surface; a Pomeranchuk instabilityVadim Oganesyan, Steven A. Kivelson, and Eduardo Fradkin,Quantum Theory of a Nematic Fermi Fluid, Phys. Rev. B 64,195109 (2001); arXiv:cond-mat/0102093.
◮ Strong coupling picture: a quantum melted stripe state. It has thesame pattern of symmetry breaking as the Pomeranchuk picture. Ithas low energy dynamical stripe fluctuations, i.e. “fluctuatingstripes”.
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Order Parameters for the Nematic State◮ Two-dimensionally translationally invariant metallic state with
spontaneously broken rotational symmetry, i.e. spontaneous“orthorhombicity”
◮ Weak coupling picture: spontaneous quadrupolar distortion of theFermi surface; a Pomeranchuk instabilityVadim Oganesyan, Steven A. Kivelson, and Eduardo Fradkin,Quantum Theory of a Nematic Fermi Fluid, Phys. Rev. B 64,195109 (2001); arXiv:cond-mat/0102093.
◮ Strong coupling picture: a quantum melted stripe state. It has thesame pattern of symmetry breaking as the Pomeranchuk picture. Ithas low energy dynamical stripe fluctuations, i.e. “fluctuatingstripes”.
◮ Order Parameter: a 2 × 2 traceless symmetric tensor, which is oddunder a π/2 rotation, e.g. extracted from the resistivity tensor
Q =ρxx − ρyy
ρxx + ρyy
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Order Parameters for the Nematic State◮ Two-dimensionally translationally invariant metallic state with
spontaneously broken rotational symmetry, i.e. spontaneous“orthorhombicity”
◮ Weak coupling picture: spontaneous quadrupolar distortion of theFermi surface; a Pomeranchuk instabilityVadim Oganesyan, Steven A. Kivelson, and Eduardo Fradkin,Quantum Theory of a Nematic Fermi Fluid, Phys. Rev. B 64,195109 (2001); arXiv:cond-mat/0102093.
◮ Strong coupling picture: a quantum melted stripe state. It has thesame pattern of symmetry breaking as the Pomeranchuk picture. Ithas low energy dynamical stripe fluctuations, i.e. “fluctuatingstripes”.
◮ Order Parameter: a 2 × 2 traceless symmetric tensor, which is oddunder a π/2 rotation, e.g. extracted from the resistivity tensor
Q =ρxx − ρyy
ρxx + ρyy
◮ Nematic charge order is fragile: charge nematic order parametercoupled to disorder ⇔ “2D random field Ising model”E. Carlson, K. Dahmen, E. Fradkin and S. Kivelson, PRL 2005
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Stripe charge order in underdoped high Tc superconductors
La2−xSrxCuO4
La2−xBaxCuO4 (non-SC)
YBa2Cu3O6+y
Neutron scattering (Tran-
quada, Mook, Keimer)
Transport (Ando)
Resonant X-ray scattering
(Abbamonte)
P. Abbamonte et al, Nature Physics 1, 155 (2005)
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Fluctuating stripe charge order and superconductivity
300
200
100
0
I 7K (
arb.
uni
ts)
-0.2 0.0 0.2
800
400
0
I 7K-I
80K
(ar
b. u
nits
)
-0.2 0.0 0.2(0.5+h,0.5,0) (0.5+h,0.5,0)
La1.86Sr0.14Cu0.988Zn0.012O4 La1.85Sr0.15CuO4
∆E = 0 ∆E = 0
∆E = 1.5 meV ∆E = 2 meV(a)
(b)
(c)
(d)
-0.2 0 0.2
400
800
1200
T = 1.5 KT = 50 K
Intensity (arb. units)100
0
-0.2 0.0 0.2
Intensity (arb. units)
S. Kivelson et al, Rev. Mod. Phys. 75, 1201 (2003) (Tranquada, Mook)
Coexistence of fluctuating stripe charge order and SC in La2−xSrxCuO4 and
YBa2Cu3O6+y
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Stripe charge order in underdoped high Tc superconductors
Underdoped La2−xSrxCuO4, x = 5% with Zn (5%)
Resonant X-ray scattering (P. Abbamonte et al, unpublished, 2007
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Charge Order induced inside a SC vortex halo
Neutron scattering in La2−xSrxCuO4 in the mixed phase (B. Lake et al)
STM in Bi2Sr2CaCu2O8+δ (S. Davis et al); J. Hoffmanet al, Science 295, 466 (2002)
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STM: Short range stripe order in Dy-Bi2Sr2CaCu2O8+δ
Kohsaka et al, Science 315, 1380 (2007);
R(~r , 150 mV ) = I (~r , +150mV )/I (~r ,−150 mV )
short range stripe order on scales ≫ ξ0(BSCCO and NaCCOC) (Kapitulnik, Davis,
Yazdani)
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Optimal Degree of Inhomogeneity in La2−xBaxCuO4
Evidence of an optimal degree of inhomogeneity in high Tc superconductors :Theantinodal (pairing) gap is largest, even though Tc is smallest, near x = 1
8in
La2−xBaxCuO4
T. Valla et al, Science 314, 1914 (2006) (ARPES)
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Why is Tc so low for La2−xBaxCuO4 at x = 1/8?
Q. Li, M. Hücker, G.D. Gu, A.M. Tsvelik and J. M. Tranquada, Phys. Rev. Lett. 99,
067001 (2007)
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Dynamical decoupling of the SC layers in La2−xBaxCuO4
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Dynamical decoupling of the SC layers in La2−xBaxCuO4
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◮ Striped state with magnetic antiphase domain walls and π shiftedsuperconductivity
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Dynamical decoupling of the SC layers in La2−xBaxCuO4
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◮ Striped state with magnetic antiphase domain walls and π shiftedsuperconductivity
◮ The c-axis Josephson coupling vanishes by symmetry in the LTTstructure
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Dynamical decoupling of the SC layers in La2−xBaxCuO4
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◮ Striped state with magnetic antiphase domain walls and π shiftedsuperconductivity
◮ The c-axis Josephson coupling vanishes by symmetry in the LTTstructure
◮ This state has 2D superconductivity but the 3D superconductivity isabsent!
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Dynamical decoupling of the SC layers in La2−xBaxCuO4
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◮ Striped state with magnetic antiphase domain walls and π shiftedsuperconductivity
◮ The c-axis Josephson coupling vanishes by symmetry in the LTTstructure
◮ This state has 2D superconductivity but the 3D superconductivity isabsent!
E. Berg, E. Fradkin. E.-A. Kim, S. A. Kivelson, V. Oganesyan, J. M.Tranquada, and S.C. Zhang, Phys. Rev. Lett. 99, 127003 (2007)
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Electron Nematic Order in 2DEG in high magnetic fields
Electron nematic phase in the 2DEG in high magnetic fields
J.P. Eisenstein et al, Phys. Rev. B 82, 394 (1999)
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Electron Nematic Order in 2DEG in high magnetic fields
K. B. Cooper et al, Phys. Rev. B 65, 241313 (2002)
◮ temperature dependent and tunable anisotropy
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Electron Nematic Order in 2DEG in high magnetic fields
K. B. Cooper et al, Phys. Rev. B 65, 241313 (2002)
◮ temperature dependent and tunable anisotropy
◮ linear I − V curves (no pinning), no noise
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Electron Nematic Order in 2DEG in high magnetic fields
K. B. Cooper et al, Phys. Rev. B 65, 241313 (2002)
◮ temperature dependent and tunable anisotropy
◮ linear I − V curves (no pinning), no noise
◮ A melted stripe state! Fradkin and Kivelson (1998), Fradkin, Kivelson,Manousakis and Nho (2000)
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Electron Nematic Order in Sr3Ru2O7 in magnetic fields
Electron nematic phase in Sr3Ru2O7
R. Borzi, S. Grigera, A. P. Mackenzie et al, Science 315, 214 (2007)
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Electron Nematic Order in Sr3Ru2O7 in magnetic fields
A nematic phase preempts a metamagnetic quantum critical point in Sr3Ru2O7
No lattice distortion has been detected
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Electron Nematic Order in high Tc superconductors
Y. Ando et al, Phys. Rev. Lett. 88, 137005 (2002)
Temperature-dependent transport anisotropy in underdoped La2−xSrxCuO4 and
YBa2Cu3O6+y
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Fluctuating Stripes as Nematic Order in underdopedYBa2Cu3O6+y
◮ Fluctuating stripe order detected in inelastic neutron scattering,Mook et al (2000)
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Fluctuating Stripes as Nematic Order in underdopedYBa2Cu3O6+y
◮ Fluctuating stripe order detected in inelastic neutron scattering,Mook et al (2000)
◮ Temperature dependence of neutron inelastic scattering peaks reveala nematic phase in YBa2Cu3O6+y , y = 4.5 and T . 150K , B.Keimer et al (2008)
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Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems
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Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems◮ Since the discovery of high Tc it has been clear that
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Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems◮ Since the discovery of high Tc it has been clear that
◮ High Tc Superconductors are never normal metals and don’t havewell defined quasiparticles in the “normal state” (linear resistivity,ARPES)
![Page 70: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/70.jpg)
Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems◮ Since the discovery of high Tc it has been clear that
◮ High Tc Superconductors are never normal metals and don’t havewell defined quasiparticles in the “normal state” (linear resistivity,ARPES)
◮ the “parent compounds” are strongly correlated Mott insulators
![Page 71: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/71.jpg)
Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems◮ Since the discovery of high Tc it has been clear that
◮ High Tc Superconductors are never normal metals and don’t havewell defined quasiparticles in the “normal state” (linear resistivity,ARPES)
◮ the “parent compounds” are strongly correlated Mott insulators◮ repulsive interactions dominate
![Page 72: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/72.jpg)
Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems◮ Since the discovery of high Tc it has been clear that
◮ High Tc Superconductors are never normal metals and don’t havewell defined quasiparticles in the “normal state” (linear resistivity,ARPES)
◮ the “parent compounds” are strongly correlated Mott insulators◮ repulsive interactions dominate◮ the quasiparticles are an ‘emergent’ low-energy property of the
superconducting state
![Page 73: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/73.jpg)
Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems◮ Since the discovery of high Tc it has been clear that
◮ High Tc Superconductors are never normal metals and don’t havewell defined quasiparticles in the “normal state” (linear resistivity,ARPES)
◮ the “parent compounds” are strongly correlated Mott insulators◮ repulsive interactions dominate◮ the quasiparticles are an ‘emergent’ low-energy property of the
superconducting state◮ whatever “the mechanism” is has to account for these facts
![Page 74: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/74.jpg)
Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems◮ Since the discovery of high Tc it has been clear that
◮ High Tc Superconductors are never normal metals and don’t havewell defined quasiparticles in the “normal state” (linear resistivity,ARPES)
◮ the “parent compounds” are strongly correlated Mott insulators◮ repulsive interactions dominate◮ the quasiparticles are an ‘emergent’ low-energy property of the
superconducting state◮ whatever “the mechanism” is has to account for these facts
Problem
![Page 75: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/75.jpg)
Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems◮ Since the discovery of high Tc it has been clear that
◮ High Tc Superconductors are never normal metals and don’t havewell defined quasiparticles in the “normal state” (linear resistivity,ARPES)
◮ the “parent compounds” are strongly correlated Mott insulators◮ repulsive interactions dominate◮ the quasiparticles are an ‘emergent’ low-energy property of the
superconducting state◮ whatever “the mechanism” is has to account for these facts
Problem
◮ BCS is so successful in conventional metals
![Page 76: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/76.jpg)
Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems◮ Since the discovery of high Tc it has been clear that
◮ High Tc Superconductors are never normal metals and don’t havewell defined quasiparticles in the “normal state” (linear resistivity,ARPES)
◮ the “parent compounds” are strongly correlated Mott insulators◮ repulsive interactions dominate◮ the quasiparticles are an ‘emergent’ low-energy property of the
superconducting state◮ whatever “the mechanism” is has to account for these facts
Problem
◮ BCS is so successful in conventional metals◮ that the term mechanism naturally evokes the idea of a weak
coupling instability
![Page 77: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/77.jpg)
Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems◮ Since the discovery of high Tc it has been clear that
◮ High Tc Superconductors are never normal metals and don’t havewell defined quasiparticles in the “normal state” (linear resistivity,ARPES)
◮ the “parent compounds” are strongly correlated Mott insulators◮ repulsive interactions dominate◮ the quasiparticles are an ‘emergent’ low-energy property of the
superconducting state◮ whatever “the mechanism” is has to account for these facts
Problem
◮ BCS is so successful in conventional metals◮ that the term mechanism naturally evokes the idea of a weak
coupling instability◮ with (write here your favorite boson) mediating an attractive
interaction between well defined quasiparticles
![Page 78: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/78.jpg)
Stripe Phases and the Mechanism of high Tc
superconductivity in Strongly Correlated Systems◮ Since the discovery of high Tc it has been clear that
◮ High Tc Superconductors are never normal metals and don’t havewell defined quasiparticles in the “normal state” (linear resistivity,ARPES)
◮ the “parent compounds” are strongly correlated Mott insulators◮ repulsive interactions dominate◮ the quasiparticles are an ‘emergent’ low-energy property of the
superconducting state◮ whatever “the mechanism” is has to account for these facts
Problem
◮ BCS is so successful in conventional metals◮ that the term mechanism naturally evokes the idea of a weak
coupling instability◮ with (write here your favorite boson) mediating an attractive
interaction between well defined quasiparticles◮ The basic assumptions of BCS theory are not satisfied in these
systems.
![Page 79: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/79.jpg)
Superconductivity in a Doped Mott Insulator
or How To Get Pairing from Repulsive Interactions
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Superconductivity in a Doped Mott Insulator
or How To Get Pairing from Repulsive Interactions
◮ Universal assumption: 2D Hubbard-like models should contain theessential physics
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Superconductivity in a Doped Mott Insulator
or How To Get Pairing from Repulsive Interactions
◮ Universal assumption: 2D Hubbard-like models should contain theessential physics
◮ “RVB” mechanism:
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Superconductivity in a Doped Mott Insulator
or How To Get Pairing from Repulsive Interactions
◮ Universal assumption: 2D Hubbard-like models should contain theessential physics
◮ “RVB” mechanism:◮ Mott insulator: spins are bound in singlet valence bonds; it is a
strongly correlated spin liquid, essentially a pre-paired insulatingstate
![Page 83: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/83.jpg)
Superconductivity in a Doped Mott Insulator
or How To Get Pairing from Repulsive Interactions
◮ Universal assumption: 2D Hubbard-like models should contain theessential physics
◮ “RVB” mechanism:◮ Mott insulator: spins are bound in singlet valence bonds; it is a
strongly correlated spin liquid, essentially a pre-paired insulatingstate
◮ spin-charge separation in the doped state leads to high Tc
superconductivity
![Page 84: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/84.jpg)
Superconductivity in a Doped Mott Insulator
or How To Get Pairing from Repulsive Interactions
◮ Universal assumption: 2D Hubbard-like models should contain theessential physics
◮ “RVB” mechanism:◮ Mott insulator: spins are bound in singlet valence bonds; it is a
strongly correlated spin liquid, essentially a pre-paired insulatingstate
◮ spin-charge separation in the doped state leads to high Tc
superconductivity
![Page 85: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/85.jpg)
Problems
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Problems
◮ there is no real evidence that the simple 2D Hubbard model favorssuperconductivity (let alone high Tc superconductivity)
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Problems
◮ there is no real evidence that the simple 2D Hubbard model favorssuperconductivity (let alone high Tc superconductivity)
◮ all evidence indicates that if anything it wants to be an insulator andto phase separate (finite size diagonalizations, various Monte Carlosimulations)
![Page 88: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/88.jpg)
Problems
◮ there is no real evidence that the simple 2D Hubbard model favorssuperconductivity (let alone high Tc superconductivity)
◮ all evidence indicates that if anything it wants to be an insulator andto phase separate (finite size diagonalizations, various Monte Carlosimulations)
◮ strong tendency for the ground states to be inhomogeneous andpossibly anisotropic
![Page 89: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/89.jpg)
Problems
◮ there is no real evidence that the simple 2D Hubbard model favorssuperconductivity (let alone high Tc superconductivity)
◮ all evidence indicates that if anything it wants to be an insulator andto phase separate (finite size diagonalizations, various Monte Carlosimulations)
◮ strong tendency for the ground states to be inhomogeneous andpossibly anisotropic
◮ no evidence (yet) for a spin liquid in 2D Hubbard-type models
![Page 90: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/90.jpg)
Problems
◮ there is no real evidence that the simple 2D Hubbard model favorssuperconductivity (let alone high Tc superconductivity)
◮ all evidence indicates that if anything it wants to be an insulator andto phase separate (finite size diagonalizations, various Monte Carlosimulations)
◮ strong tendency for the ground states to be inhomogeneous andpossibly anisotropic
◮ no evidence (yet) for a spin liquid in 2D Hubbard-type models
![Page 91: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/91.jpg)
Why an Inhomogeneous State is Good for high Tc SC
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Why an Inhomogeneous State is Good for high Tc SC◮ An “inhomogeneity induced pairing” mechanism of high temperature
superconductivity in which the pairing of electrons originates directlyfrom strong repulsive interactions.
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Why an Inhomogeneous State is Good for high Tc SC◮ An “inhomogeneity induced pairing” mechanism of high temperature
superconductivity in which the pairing of electrons originates directlyfrom strong repulsive interactions.
◮ Repulsive interactions lead to local superconductivity on ‘mesoscalestructures’
![Page 94: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/94.jpg)
Why an Inhomogeneous State is Good for high Tc SC◮ An “inhomogeneity induced pairing” mechanism of high temperature
superconductivity in which the pairing of electrons originates directlyfrom strong repulsive interactions.
◮ Repulsive interactions lead to local superconductivity on ‘mesoscalestructures’
◮ The strength of this pairing tendency decreases as the size of thestructures increases above an optimal size
![Page 95: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/95.jpg)
Why an Inhomogeneous State is Good for high Tc SC◮ An “inhomogeneity induced pairing” mechanism of high temperature
superconductivity in which the pairing of electrons originates directlyfrom strong repulsive interactions.
◮ Repulsive interactions lead to local superconductivity on ‘mesoscalestructures’
◮ The strength of this pairing tendency decreases as the size of thestructures increases above an optimal size
◮ The physics responsible for the pairing within a structure ⇒Coulomb frustrated phase separation ⇒ mesoscale electronicstructures
![Page 96: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/96.jpg)
Why an Inhomogeneous State is Good for high Tc SC◮ An “inhomogeneity induced pairing” mechanism of high temperature
superconductivity in which the pairing of electrons originates directlyfrom strong repulsive interactions.
◮ Repulsive interactions lead to local superconductivity on ‘mesoscalestructures’
◮ The strength of this pairing tendency decreases as the size of thestructures increases above an optimal size
◮ The physics responsible for the pairing within a structure ⇒Coulomb frustrated phase separation ⇒ mesoscale electronicstructures
◮ Strong local pairing does not guarantee a large critical temperature
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Why an Inhomogeneous State is Good for high Tc SC◮ An “inhomogeneity induced pairing” mechanism of high temperature
superconductivity in which the pairing of electrons originates directlyfrom strong repulsive interactions.
◮ Repulsive interactions lead to local superconductivity on ‘mesoscalestructures’
◮ The strength of this pairing tendency decreases as the size of thestructures increases above an optimal size
◮ The physics responsible for the pairing within a structure ⇒Coulomb frustrated phase separation ⇒ mesoscale electronicstructures
◮ Strong local pairing does not guarantee a large critical temperature◮ In an isolated system, the phase ordering (condensation) temperature
is suppressed by phase fluctuations, often to T = 0
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Why an Inhomogeneous State is Good for high Tc SC◮ An “inhomogeneity induced pairing” mechanism of high temperature
superconductivity in which the pairing of electrons originates directlyfrom strong repulsive interactions.
◮ Repulsive interactions lead to local superconductivity on ‘mesoscalestructures’
◮ The strength of this pairing tendency decreases as the size of thestructures increases above an optimal size
◮ The physics responsible for the pairing within a structure ⇒Coulomb frustrated phase separation ⇒ mesoscale electronicstructures
◮ Strong local pairing does not guarantee a large critical temperature◮ In an isolated system, the phase ordering (condensation) temperature
is suppressed by phase fluctuations, often to T = 0◮ The highest possible Tc is obtained with an intermediate degree of
inhomogeneity
![Page 99: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/99.jpg)
Why an Inhomogeneous State is Good for high Tc SC◮ An “inhomogeneity induced pairing” mechanism of high temperature
superconductivity in which the pairing of electrons originates directlyfrom strong repulsive interactions.
◮ Repulsive interactions lead to local superconductivity on ‘mesoscalestructures’
◮ The strength of this pairing tendency decreases as the size of thestructures increases above an optimal size
◮ The physics responsible for the pairing within a structure ⇒Coulomb frustrated phase separation ⇒ mesoscale electronicstructures
◮ Strong local pairing does not guarantee a large critical temperature◮ In an isolated system, the phase ordering (condensation) temperature
is suppressed by phase fluctuations, often to T = 0◮ The highest possible Tc is obtained with an intermediate degree of
inhomogeneity◮ The optimal Tc always occurs at a point of crossover from a pairing
dominated regime when the degree of inhomogeneity is suboptimal,to a phase ordering regime with a pseudo-gap when the system is too‘granular’
![Page 100: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/100.jpg)
Why an Inhomogeneous State is Good for high Tc SC◮ An “inhomogeneity induced pairing” mechanism of high temperature
superconductivity in which the pairing of electrons originates directlyfrom strong repulsive interactions.
◮ Repulsive interactions lead to local superconductivity on ‘mesoscalestructures’
◮ The strength of this pairing tendency decreases as the size of thestructures increases above an optimal size
◮ The physics responsible for the pairing within a structure ⇒Coulomb frustrated phase separation ⇒ mesoscale electronicstructures
◮ Strong local pairing does not guarantee a large critical temperature◮ In an isolated system, the phase ordering (condensation) temperature
is suppressed by phase fluctuations, often to T = 0◮ The highest possible Tc is obtained with an intermediate degree of
inhomogeneity◮ The optimal Tc always occurs at a point of crossover from a pairing
dominated regime when the degree of inhomogeneity is suboptimal,to a phase ordering regime with a pseudo-gap when the system is too‘granular’
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Striped Hubbard Model: A Cartoon of the StronglyCorrelated Stripe Phase
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Striped Hubbard Model: A Cartoon of the StronglyCorrelated Stripe Phase
H = −
X
<~r ,~r ′>,σ
t~r ,~r ′
h
c†
~r,σc~r ′,σ + h.c.
i
+
X
~r ,σ
»
ǫ~r c†
~r ,σc~r ,σ +
U
2c†
~r,σc†
~r ,−σc~r ,−σc~r ,σ
–
tttt
t′t′ δtδtδt
εε −ε−ε
A B
Enrico Arrigoni, Eduardo Fradkin and Steven A. Kivelson,Mechanism of High Temperature Superconductivity in a striped Hubbard ModelPhys. Rev. B 69, 214519 (2004); arXiv:cond-mat/0309572.
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Physics of the 2-leg ladder
t
t′
U
V
E
p
EF
pF1−pF1
pF2−pF2
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Physics of the 2-leg ladder
t
t′
U
V
E
p
EF
pF1−pF1
pF2−pF2
◮ For U = V = 0 there are two bands of states
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Physics of the 2-leg ladder
t
t′
U
V
E
p
EF
pF1−pF1
pF2−pF2
◮ For U = V = 0 there are two bands of states
◮ The bands have different Fermi wave vectors, pF1 6= pF2
![Page 106: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/106.jpg)
Physics of the 2-leg ladder
t
t′
U
V
E
p
EF
pF1−pF1
pF2−pF2
◮ For U = V = 0 there are two bands of states
◮ The bands have different Fermi wave vectors, pF1 6= pF2
◮ The only allowed scattering processes involve an even number ofelectrons (momentum conservation)
![Page 107: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/107.jpg)
Physics of the 2-leg ladder
t
t′
U
V
E
p
EF
pF1−pF1
pF2−pF2
◮ For U = V = 0 there are two bands of states
◮ The bands have different Fermi wave vectors, pF1 6= pF2
◮ The only allowed scattering processes involve an even number ofelectrons (momentum conservation)
◮ The coupling of CDW fluctuations with Q1 = 2pF1 6= Q2 = 2pF2 issuppressed
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Physics of the 2-leg ladder
t
t′
U
V
E
p
EF
pF1−pF1
pF2−pF2
◮ For U = V = 0 there are two bands of states
◮ The bands have different Fermi wave vectors, pF1 6= pF2
◮ The only allowed scattering processes involve an even number ofelectrons (momentum conservation)
◮ The coupling of CDW fluctuations with Q1 = 2pF1 6= Q2 = 2pF2 issuppressed
![Page 109: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/109.jpg)
Why is there a Spin Gap
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Why is there a Spin Gap
◮ Scattering of electron pairs with zero center of mass momentumfrom one system to the other is peturbatively relevant
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Why is there a Spin Gap
◮ Scattering of electron pairs with zero center of mass momentumfrom one system to the other is peturbatively relevant
◮ The electrons can gain zero-point energy by delocalizing between thetwo bands
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Why is there a Spin Gap
◮ Scattering of electron pairs with zero center of mass momentumfrom one system to the other is peturbatively relevant
◮ The electrons can gain zero-point energy by delocalizing between thetwo bands
◮ The electrons need to pair, which may cost some energy
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Why is there a Spin Gap
◮ Scattering of electron pairs with zero center of mass momentumfrom one system to the other is peturbatively relevant
◮ The electrons can gain zero-point energy by delocalizing between thetwo bands
◮ The electrons need to pair, which may cost some energy
◮ When the energy gained by delocalizing between the two bandsexceeds the energy cost of pairing, the system is driven to a spin-gapphase
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Why is there a Spin Gap
◮ Scattering of electron pairs with zero center of mass momentumfrom one system to the other is peturbatively relevant
◮ The electrons can gain zero-point energy by delocalizing between thetwo bands
◮ The electrons need to pair, which may cost some energy
◮ When the energy gained by delocalizing between the two bandsexceeds the energy cost of pairing, the system is driven to a spin-gapphase
◮ This is corroborated by extensive DMRG numerical calculations thatfind a spin gap for a broad range of parameters (doping and couplingconstants) of doped 2-leg and 3-leg ladders (White, Noack andScalapino), and analytically by bosonization methods Giamarchi andSchulz; Wu, Liu and Fradkin
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What is it known about the 2-leg ladder
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What is it known about the 2-leg ladder
◮ At x = 0 there is a unique fully gapped ground state (“C0S0”); forU ≫ t, ∆s ∼ J/2
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What is it known about the 2-leg ladder
◮ At x = 0 there is a unique fully gapped ground state (“C0S0”); forU ≫ t, ∆s ∼ J/2
◮ For 0 < x < xc ∼ 0.3, Luther-Emery liquid: no charge gap and largespin gap (“C1S0”); spin gap ∆s ↓ as x ↑, and ∆s → 0 as x → xc
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What is it known about the 2-leg ladder
◮ At x = 0 there is a unique fully gapped ground state (“C0S0”); forU ≫ t, ∆s ∼ J/2
◮ For 0 < x < xc ∼ 0.3, Luther-Emery liquid: no charge gap and largespin gap (“C1S0”); spin gap ∆s ↓ as x ↑, and ∆s → 0 as x → xc
◮ Effective Hamiltonian for the charge degrees of freedom
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What is it known about the 2-leg ladder
◮ At x = 0 there is a unique fully gapped ground state (“C0S0”); forU ≫ t, ∆s ∼ J/2
◮ For 0 < x < xc ∼ 0.3, Luther-Emery liquid: no charge gap and largespin gap (“C1S0”); spin gap ∆s ↓ as x ↑, and ∆s → 0 as x → xc
◮ Effective Hamiltonian for the charge degrees of freedom
H =
∫dy
vc
2
[K (∂yθ)2 +
1
K(∂xφ)2
]+ . . .
φ: CDW phase field; θ: SC phase field; [φ(y ′), ∂yθ(y)] = iδ(y − y ′)
![Page 120: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/120.jpg)
What is it known about the 2-leg ladder
◮ At x = 0 there is a unique fully gapped ground state (“C0S0”); forU ≫ t, ∆s ∼ J/2
◮ For 0 < x < xc ∼ 0.3, Luther-Emery liquid: no charge gap and largespin gap (“C1S0”); spin gap ∆s ↓ as x ↑, and ∆s → 0 as x → xc
◮ Effective Hamiltonian for the charge degrees of freedom
H =
∫dy
vc
2
[K (∂yθ)2 +
1
K(∂xφ)2
]+ . . .
φ: CDW phase field; θ: SC phase field; [φ(y ′), ∂yθ(y)] = iδ(y − y ′)
◮ x-dependence of ∆s , K , vc , and µ depends on t ′/t and U/t
![Page 121: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/121.jpg)
What is it known about the 2-leg ladder
◮ At x = 0 there is a unique fully gapped ground state (“C0S0”); forU ≫ t, ∆s ∼ J/2
◮ For 0 < x < xc ∼ 0.3, Luther-Emery liquid: no charge gap and largespin gap (“C1S0”); spin gap ∆s ↓ as x ↑, and ∆s → 0 as x → xc
◮ Effective Hamiltonian for the charge degrees of freedom
H =
∫dy
vc
2
[K (∂yθ)2 +
1
K(∂xφ)2
]+ . . .
φ: CDW phase field; θ: SC phase field; [φ(y ′), ∂yθ(y)] = iδ(y − y ′)
◮ x-dependence of ∆s , K , vc , and µ depends on t ′/t and U/t
◮ . . . represent cosine potentials: Mott gap ∆M at x = 0
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What is it known about the 2-leg ladder
◮ At x = 0 there is a unique fully gapped ground state (“C0S0”); forU ≫ t, ∆s ∼ J/2
◮ For 0 < x < xc ∼ 0.3, Luther-Emery liquid: no charge gap and largespin gap (“C1S0”); spin gap ∆s ↓ as x ↑, and ∆s → 0 as x → xc
◮ Effective Hamiltonian for the charge degrees of freedom
H =
∫dy
vc
2
[K (∂yθ)2 +
1
K(∂xφ)2
]+ . . .
φ: CDW phase field; θ: SC phase field; [φ(y ′), ∂yθ(y)] = iδ(y − y ′)
◮ x-dependence of ∆s , K , vc , and µ depends on t ′/t and U/t
◮ . . . represent cosine potentials: Mott gap ∆M at x = 0
◮ K → 2 as x → 0; K ∼ 1 for x ∼ 0.1, and K ∼ 1/2 for x ∼ xc
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What is it known about the 2-leg ladder
◮ At x = 0 there is a unique fully gapped ground state (“C0S0”); forU ≫ t, ∆s ∼ J/2
◮ For 0 < x < xc ∼ 0.3, Luther-Emery liquid: no charge gap and largespin gap (“C1S0”); spin gap ∆s ↓ as x ↑, and ∆s → 0 as x → xc
◮ Effective Hamiltonian for the charge degrees of freedom
H =
∫dy
vc
2
[K (∂yθ)2 +
1
K(∂xφ)2
]+ . . .
φ: CDW phase field; θ: SC phase field; [φ(y ′), ∂yθ(y)] = iδ(y − y ′)
◮ x-dependence of ∆s , K , vc , and µ depends on t ′/t and U/t
◮ . . . represent cosine potentials: Mott gap ∆M at x = 0
◮ K → 2 as x → 0; K ∼ 1 for x ∼ 0.1, and K ∼ 1/2 for x ∼ xc
◮ χSC ∼ ∆s/T 2−K−1
χCDW ∼ ∆s/T 2−K
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What is it known about the 2-leg ladder
◮ At x = 0 there is a unique fully gapped ground state (“C0S0”); forU ≫ t, ∆s ∼ J/2
◮ For 0 < x < xc ∼ 0.3, Luther-Emery liquid: no charge gap and largespin gap (“C1S0”); spin gap ∆s ↓ as x ↑, and ∆s → 0 as x → xc
◮ Effective Hamiltonian for the charge degrees of freedom
H =
∫dy
vc
2
[K (∂yθ)2 +
1
K(∂xφ)2
]+ . . .
φ: CDW phase field; θ: SC phase field; [φ(y ′), ∂yθ(y)] = iδ(y − y ′)
◮ x-dependence of ∆s , K , vc , and µ depends on t ′/t and U/t
◮ . . . represent cosine potentials: Mott gap ∆M at x = 0
◮ K → 2 as x → 0; K ∼ 1 for x ∼ 0.1, and K ∼ 1/2 for x ∼ xc
◮ χSC ∼ ∆s/T 2−K−1
χCDW ∼ ∆s/T 2−K
◮ χCDW(T ) → ∞ and χSC(T ) → ∞ for 0 < x < xc
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What is it known about the 2-leg ladder
◮ At x = 0 there is a unique fully gapped ground state (“C0S0”); forU ≫ t, ∆s ∼ J/2
◮ For 0 < x < xc ∼ 0.3, Luther-Emery liquid: no charge gap and largespin gap (“C1S0”); spin gap ∆s ↓ as x ↑, and ∆s → 0 as x → xc
◮ Effective Hamiltonian for the charge degrees of freedom
H =
∫dy
vc
2
[K (∂yθ)2 +
1
K(∂xφ)2
]+ . . .
φ: CDW phase field; θ: SC phase field; [φ(y ′), ∂yθ(y)] = iδ(y − y ′)
◮ x-dependence of ∆s , K , vc , and µ depends on t ′/t and U/t
◮ . . . represent cosine potentials: Mott gap ∆M at x = 0
◮ K → 2 as x → 0; K ∼ 1 for x ∼ 0.1, and K ∼ 1/2 for x ∼ xc
◮ χSC ∼ ∆s/T 2−K−1
χCDW ∼ ∆s/T 2−K
◮ χCDW(T ) → ∞ and χSC(T ) → ∞ for 0 < x < xc
◮ For x . 0.1, χSC ≫ χCDW!
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What is it known about the 2-leg ladder
◮ At x = 0 there is a unique fully gapped ground state (“C0S0”); forU ≫ t, ∆s ∼ J/2
◮ For 0 < x < xc ∼ 0.3, Luther-Emery liquid: no charge gap and largespin gap (“C1S0”); spin gap ∆s ↓ as x ↑, and ∆s → 0 as x → xc
◮ Effective Hamiltonian for the charge degrees of freedom
H =
∫dy
vc
2
[K (∂yθ)2 +
1
K(∂xφ)2
]+ . . .
φ: CDW phase field; θ: SC phase field; [φ(y ′), ∂yθ(y)] = iδ(y − y ′)
◮ x-dependence of ∆s , K , vc , and µ depends on t ′/t and U/t
◮ . . . represent cosine potentials: Mott gap ∆M at x = 0
◮ K → 2 as x → 0; K ∼ 1 for x ∼ 0.1, and K ∼ 1/2 for x ∼ xc
◮ χSC ∼ ∆s/T 2−K−1
χCDW ∼ ∆s/T 2−K
◮ χCDW(T ) → ∞ and χSC(T ) → ∞ for 0 < x < xc
◮ For x . 0.1, χSC ≫ χCDW!
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Effects of Inter-ladder Couplings
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Effects of Inter-ladder Couplings
◮ In the Luther-Emery phase, 0 < x < xc , there is a spin gap andsingle particle tunneling is irrelevant
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Effects of Inter-ladder Couplings
◮ In the Luther-Emery phase, 0 < x < xc , there is a spin gap andsingle particle tunneling is irrelevant
◮ Second order processes in δt:
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Effects of Inter-ladder Couplings
◮ In the Luther-Emery phase, 0 < x < xc , there is a spin gap andsingle particle tunneling is irrelevant
◮ Second order processes in δt:◮ marginal (and small) forward scattering inter-ladder interactions
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Effects of Inter-ladder Couplings
◮ In the Luther-Emery phase, 0 < x < xc , there is a spin gap andsingle particle tunneling is irrelevant
◮ Second order processes in δt:◮ marginal (and small) forward scattering inter-ladder interactions◮ Josephson couplings, possibly relevant
![Page 132: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/132.jpg)
Effects of Inter-ladder Couplings
◮ In the Luther-Emery phase, 0 < x < xc , there is a spin gap andsingle particle tunneling is irrelevant
◮ Second order processes in δt:◮ marginal (and small) forward scattering inter-ladder interactions◮ Josephson couplings, possibly relevant◮ CDW couplings, possibly relevant
![Page 133: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/133.jpg)
Effects of Inter-ladder Couplings
◮ In the Luther-Emery phase, 0 < x < xc , there is a spin gap andsingle particle tunneling is irrelevant
◮ Second order processes in δt:◮ marginal (and small) forward scattering inter-ladder interactions◮ Josephson couplings, possibly relevant◮ CDW couplings, possibly relevant
◮ Relevant Perturbations
![Page 134: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/134.jpg)
Effects of Inter-ladder Couplings
◮ In the Luther-Emery phase, 0 < x < xc , there is a spin gap andsingle particle tunneling is irrelevant
◮ Second order processes in δt:◮ marginal (and small) forward scattering inter-ladder interactions◮ Josephson couplings, possibly relevant◮ CDW couplings, possibly relevant
◮ Relevant Perturbations
H ′ = −∑
J
∫dy
[J cos
(√2π∆θJ)
)+ V cos
(∆PJy +
√2π∆φJ)
)]
J: ladder index; PJ = 2πxJ , ∆φJ = φJ+1 − φJ , etc.
![Page 135: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/135.jpg)
Effects of Inter-ladder Couplings
◮ In the Luther-Emery phase, 0 < x < xc , there is a spin gap andsingle particle tunneling is irrelevant
◮ Second order processes in δt:◮ marginal (and small) forward scattering inter-ladder interactions◮ Josephson couplings, possibly relevant◮ CDW couplings, possibly relevant
◮ Relevant Perturbations
H ′ = −∑
J
∫dy
[J cos
(√2π∆θJ)
)+ V cos
(∆PJy +
√2π∆φJ)
)]
J: ladder index; PJ = 2πxJ , ∆φJ = φJ+1 − φJ , etc.
◮ J and V are effective couplings which must be computed frommicroscopics
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Effects of Inter-ladder Couplings
◮ In the Luther-Emery phase, 0 < x < xc , there is a spin gap andsingle particle tunneling is irrelevant
◮ Second order processes in δt:◮ marginal (and small) forward scattering inter-ladder interactions◮ Josephson couplings, possibly relevant◮ CDW couplings, possibly relevant
◮ Relevant Perturbations
H ′ = −∑
J
∫dy
[J cos
(√2π∆θJ)
)+ V cos
(∆PJy +
√2π∆φJ)
)]
J: ladder index; PJ = 2πxJ , ∆φJ = φJ+1 − φJ , etc.
◮ J and V are effective couplings which must be computed frommicroscopics
◮ Estimate: J ≈ V ∝ (δt)2/J
![Page 137: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/137.jpg)
Effects of Inter-ladder Couplings
◮ In the Luther-Emery phase, 0 < x < xc , there is a spin gap andsingle particle tunneling is irrelevant
◮ Second order processes in δt:◮ marginal (and small) forward scattering inter-ladder interactions◮ Josephson couplings, possibly relevant◮ CDW couplings, possibly relevant
◮ Relevant Perturbations
H ′ = −∑
J
∫dy
[J cos
(√2π∆θJ)
)+ V cos
(∆PJy +
√2π∆φJ)
)]
J: ladder index; PJ = 2πxJ , ∆φJ = φJ+1 − φJ , etc.
◮ J and V are effective couplings which must be computed frommicroscopics
◮ Estimate: J ≈ V ∝ (δt)2/J
![Page 138: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/138.jpg)
Period 2 works for x ≪ 1
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Period 2 works for x ≪ 1
◮ If all the ladders are equivalent, a period 2 stripe ordered or columnstate
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Period 2 works for x ≪ 1
◮ If all the ladders are equivalent, a period 2 stripe ordered or columnstate
◮ For an isolated ladder Tc = 0
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Period 2 works for x ≪ 1
◮ If all the ladders are equivalent, a period 2 stripe ordered or columnstate
◮ For an isolated ladder Tc = 0
◮ J 6= 0 and V 6= 0, TC > 0
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Period 2 works for x ≪ 1
◮ If all the ladders are equivalent, a period 2 stripe ordered or columnstate
◮ For an isolated ladder Tc = 0
◮ J 6= 0 and V 6= 0, TC > 0
◮ For x . 0.1 CDW couplings are irrelevant (1 < K < 2): Inter-ladderJosephson coupling leads to a superconducting state in a restrictedrange of small x with rather low Tc .
2JχSC(Tc) = 1
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Period 2 works for x ≪ 1
◮ If all the ladders are equivalent, a period 2 stripe ordered or columnstate
◮ For an isolated ladder Tc = 0
◮ J 6= 0 and V 6= 0, TC > 0
◮ For x . 0.1 CDW couplings are irrelevant (1 < K < 2): Inter-ladderJosephson coupling leads to a superconducting state in a restrictedrange of small x with rather low Tc .
2JχSC(Tc) = 1
◮ Tc ∝ δt x
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Period 2 works for x ≪ 1
◮ If all the ladders are equivalent, a period 2 stripe ordered or columnstate
◮ For an isolated ladder Tc = 0
◮ J 6= 0 and V 6= 0, TC > 0
◮ For x . 0.1 CDW couplings are irrelevant (1 < K < 2): Inter-ladderJosephson coupling leads to a superconducting state in a restrictedrange of small x with rather low Tc .
2JχSC(Tc) = 1
◮ Tc ∝ δt x
◮ For larger x , K < 1 and χCDW is more strongly divergent than χSC
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Period 2 works for x ≪ 1
◮ If all the ladders are equivalent, a period 2 stripe ordered or columnstate
◮ For an isolated ladder Tc = 0
◮ J 6= 0 and V 6= 0, TC > 0
◮ For x . 0.1 CDW couplings are irrelevant (1 < K < 2): Inter-ladderJosephson coupling leads to a superconducting state in a restrictedrange of small x with rather low Tc .
2JχSC(Tc) = 1
◮ Tc ∝ δt x
◮ For larger x , K < 1 and χCDW is more strongly divergent than χSC
◮ CDW couplings become more relevant ⇒ Insulating,incommensurate CDW state with ordering wave number P = 2πx .
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Period 2 works for x ≪ 1
◮ If all the ladders are equivalent, a period 2 stripe ordered or columnstate
◮ For an isolated ladder Tc = 0
◮ J 6= 0 and V 6= 0, TC > 0
◮ For x . 0.1 CDW couplings are irrelevant (1 < K < 2): Inter-ladderJosephson coupling leads to a superconducting state in a restrictedrange of small x with rather low Tc .
2JχSC(Tc) = 1
◮ Tc ∝ δt x
◮ For larger x , K < 1 and χCDW is more strongly divergent than χSC
◮ CDW couplings become more relevant ⇒ Insulating,incommensurate CDW state with ordering wave number P = 2πx .
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Why Period 4 works even better!
![Page 148: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/148.jpg)
Why Period 4 works even better!◮ Consider an alternating array of inequivalent A and B type ladders in
the LE regime
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Why Period 4 works even better!◮ Consider an alternating array of inequivalent A and B type ladders in
the LE regime◮ SC Tc :
(2J )2χASC(Tc)χ
BSC(Tc) = 1
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Why Period 4 works even better!◮ Consider an alternating array of inequivalent A and B type ladders in
the LE regime◮ SC Tc :
(2J )2χASC(Tc)χ
BSC(Tc) = 1
◮ CDW Tc :(2V)2χA
CDW(P , Tc )χBCDW(P , Tc) = 1
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Why Period 4 works even better!◮ Consider an alternating array of inequivalent A and B type ladders in
the LE regime◮ SC Tc :
(2J )2χASC(Tc)χ
BSC(Tc) = 1
◮ CDW Tc :(2V)2χA
CDW(P , Tc )χBCDW(P , Tc) = 1
◮ 2D CDW order is greatly suppressed due to the mismatch betweenordering vectors, PA and PB , on neighboring ladders
![Page 152: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/152.jpg)
Why Period 4 works even better!◮ Consider an alternating array of inequivalent A and B type ladders in
the LE regime◮ SC Tc :
(2J )2χASC(Tc)χ
BSC(Tc) = 1
◮ CDW Tc :(2V)2χA
CDW(P , Tc )χBCDW(P , Tc) = 1
◮ 2D CDW order is greatly suppressed due to the mismatch betweenordering vectors, PA and PB , on neighboring ladders
◮ For inequivalent ladders SC beats CDW if
![Page 153: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/153.jpg)
Why Period 4 works even better!◮ Consider an alternating array of inequivalent A and B type ladders in
the LE regime◮ SC Tc :
(2J )2χASC(Tc)χ
BSC(Tc) = 1
◮ CDW Tc :(2V)2χA
CDW(P , Tc )χBCDW(P , Tc) = 1
◮ 2D CDW order is greatly suppressed due to the mismatch betweenordering vectors, PA and PB , on neighboring ladders
◮ For inequivalent ladders SC beats CDW if
2 > K−1A + K−1
B − KA; 2 > K−1A + K−1
B − KB
![Page 154: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/154.jpg)
Why Period 4 works even better!◮ Consider an alternating array of inequivalent A and B type ladders in
the LE regime◮ SC Tc :
(2J )2χASC(Tc)χ
BSC(Tc) = 1
◮ CDW Tc :(2V)2χA
CDW(P , Tc )χBCDW(P , Tc) = 1
◮ 2D CDW order is greatly suppressed due to the mismatch betweenordering vectors, PA and PB , on neighboring ladders
◮ For inequivalent ladders SC beats CDW if
2 > K−1A + K−1
B − KA; 2 > K−1A + K−1
B − KB
◮
Tc ∼ ∆s
( JW
)α
; α =2KAKB
[4KAKB − KA − KB ]
![Page 155: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/155.jpg)
Why Period 4 works even better!◮ Consider an alternating array of inequivalent A and B type ladders in
the LE regime◮ SC Tc :
(2J )2χASC(Tc)χ
BSC(Tc) = 1
◮ CDW Tc :(2V)2χA
CDW(P , Tc )χBCDW(P , Tc) = 1
◮ 2D CDW order is greatly suppressed due to the mismatch betweenordering vectors, PA and PB , on neighboring ladders
◮ For inequivalent ladders SC beats CDW if
2 > K−1A + K−1
B − KA; 2 > K−1A + K−1
B − KB
◮
Tc ∼ ∆s
( JW
)α
; α =2KAKB
[4KAKB − KA − KB ]
◮ J ∼ δt2/J and W ∼ J;Tc is (power law) small for small J ! (α ∼ 1).
![Page 156: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/156.jpg)
How reliable are these estimates?
![Page 157: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/157.jpg)
How reliable are these estimates?
◮ This is a mean-field estimate for Tc and it is an upper bound to theactual Tc .
![Page 158: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/158.jpg)
How reliable are these estimates?
◮ This is a mean-field estimate for Tc and it is an upper bound to theactual Tc .
◮ Tc should be suppressed by phase fluctuations by up to a factor of 2.
![Page 159: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/159.jpg)
How reliable are these estimates?
◮ This is a mean-field estimate for Tc and it is an upper bound to theactual Tc .
◮ Tc should be suppressed by phase fluctuations by up to a factor of 2.
◮ Indeed, perturbative RG studies for small J yield the same power
law dependence. This result is asymptotically exact for J << W .
![Page 160: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/160.jpg)
How reliable are these estimates?
◮ This is a mean-field estimate for Tc and it is an upper bound to theactual Tc .
◮ Tc should be suppressed by phase fluctuations by up to a factor of 2.
◮ Indeed, perturbative RG studies for small J yield the same power
law dependence. This result is asymptotically exact for J << W .
◮ Since Tc is a smooth function of δt/J , it is reasonable toextrapolate for δt ∼ J .
![Page 161: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/161.jpg)
How reliable are these estimates?
◮ This is a mean-field estimate for Tc and it is an upper bound to theactual Tc .
◮ Tc should be suppressed by phase fluctuations by up to a factor of 2.
◮ Indeed, perturbative RG studies for small J yield the same power
law dependence. This result is asymptotically exact for J << W .
◮ Since Tc is a smooth function of δt/J , it is reasonable toextrapolate for δt ∼ J .
◮ ⇒ Tmaxc ∝ ∆s ⇒high Tc .
![Page 162: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/162.jpg)
How reliable are these estimates?
◮ This is a mean-field estimate for Tc and it is an upper bound to theactual Tc .
◮ Tc should be suppressed by phase fluctuations by up to a factor of 2.
◮ Indeed, perturbative RG studies for small J yield the same power
law dependence. This result is asymptotically exact for J << W .
◮ Since Tc is a smooth function of δt/J , it is reasonable toextrapolate for δt ∼ J .
◮ ⇒ Tmaxc ∝ ∆s ⇒high Tc .
◮ This is in contrast to the exponentially small Tc as obtained in aBCS-like mechanism.
![Page 163: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/163.jpg)
Schematic Phase Diagram for Period 2 and Period 4
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Schematic Phase Diagram for Period 2 and Period 4
Tc
∆s(x)
0
SC CDW
xxcxc(2) xc(4)
J
2
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Schematic Phase Diagram for Period 2 and Period 4
Tc
∆s(x)
0
SC CDW
xxcxc(2) xc(4)
J
2
◮ The broken line is the spin gap ∆s(x) as a function of doping x
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Schematic Phase Diagram for Period 2 and Period 4
Tc
∆s(x)
0
SC CDW
xxcxc(2) xc(4)
J
2
◮ The broken line is the spin gap ∆s(x) as a function of doping x
◮ xc(2) and xc(4) indicates the SC-CDW quantum phase transition forperiod 2 and period 4
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Schematic Phase Diagram for Period 2 and Period 4
Tc
∆s(x)
0
SC CDW
xxcxc(2) xc(4)
J
2
◮ The broken line is the spin gap ∆s(x) as a function of doping x
◮ xc(2) and xc(4) indicates the SC-CDW quantum phase transition forperiod 2 and period 4
◮ For x & xc the isolated ladders do not have a spin gap; in thisregime the physics is different involving low-energy spin fluctuations
![Page 168: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/168.jpg)
Open Questions
![Page 169: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/169.jpg)
Open Questions
◮ a period 2 modulation can produce superconductivity with arelatively low Tc in a restricted doping range
![Page 170: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/170.jpg)
Open Questions
◮ a period 2 modulation can produce superconductivity with arelatively low Tc in a restricted doping range
◮ a period 4 modulation produces higher Tc ’s on a broader range ofdoping
![Page 171: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/171.jpg)
Open Questions
◮ a period 2 modulation can produce superconductivity with arelatively low Tc in a restricted doping range
◮ a period 4 modulation produces higher Tc ’s on a broader range ofdoping
◮ Tc is only power-law small, with α ∼ 1
![Page 172: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/172.jpg)
Open Questions
◮ a period 2 modulation can produce superconductivity with arelatively low Tc in a restricted doping range
◮ a period 4 modulation produces higher Tc ’s on a broader range ofdoping
◮ Tc is only power-law small, with α ∼ 1
◮ no exponential suppression of Tc ⇒ “high Tc ”
![Page 173: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/173.jpg)
Open Questions
◮ a period 2 modulation can produce superconductivity with arelatively low Tc in a restricted doping range
◮ a period 4 modulation produces higher Tc ’s on a broader range ofdoping
◮ Tc is only power-law small, with α ∼ 1
◮ no exponential suppression of Tc ⇒ “high Tc ”
◮ This model is cartoon of the symmetry breaking of stripe (smectic)state
![Page 174: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/174.jpg)
Open Questions
◮ a period 2 modulation can produce superconductivity with arelatively low Tc in a restricted doping range
◮ a period 4 modulation produces higher Tc ’s on a broader range ofdoping
◮ Tc is only power-law small, with α ∼ 1
◮ no exponential suppression of Tc ⇒ “high Tc ”
◮ This model is cartoon of the symmetry breaking of stripe (smectic)state
◮ It has a large spin gap and it does not have low-energy spinfluctuations
![Page 175: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/175.jpg)
Open Questions
◮ a period 2 modulation can produce superconductivity with arelatively low Tc in a restricted doping range
◮ a period 4 modulation produces higher Tc ’s on a broader range ofdoping
◮ Tc is only power-law small, with α ∼ 1
◮ no exponential suppression of Tc ⇒ “high Tc ”
◮ This model is cartoon of the symmetry breaking of stripe (smectic)state
◮ It has a large spin gap and it does not have low-energy spinfluctuations
◮ the order-parameter is d-wave like: it changes sign under π/2rotations
![Page 176: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/176.jpg)
Open Questions
◮ a period 2 modulation can produce superconductivity with arelatively low Tc in a restricted doping range
◮ a period 4 modulation produces higher Tc ’s on a broader range ofdoping
◮ Tc is only power-law small, with α ∼ 1
◮ no exponential suppression of Tc ⇒ “high Tc ”
◮ This model is cartoon of the symmetry breaking of stripe (smectic)state
◮ It has a large spin gap and it does not have low-energy spinfluctuations
◮ the order-parameter is d-wave like: it changes sign under π/2rotations
◮ It does not have nodal fermionic excitations
![Page 177: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/177.jpg)
Open Questions
◮ a period 2 modulation can produce superconductivity with arelatively low Tc in a restricted doping range
◮ a period 4 modulation produces higher Tc ’s on a broader range ofdoping
◮ Tc is only power-law small, with α ∼ 1
◮ no exponential suppression of Tc ⇒ “high Tc ”
◮ This model is cartoon of the symmetry breaking of stripe (smectic)state
◮ It has a large spin gap and it does not have low-energy spinfluctuations
◮ the order-parameter is d-wave like: it changes sign under π/2rotations
◮ It does not have nodal fermionic excitations
◮ Nodal fermions may appear upon a (Lifshitz) transition at larger x
![Page 178: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/178.jpg)
Punch Line
![Page 179: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/179.jpg)
Punch Line
◮ Long held belief: charge order competes and suppressessuperconductivity
![Page 180: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/180.jpg)
Punch Line
◮ Long held belief: charge order competes and suppressessuperconductivity
◮ Electronic liquid crystal phases, not only can coexist withsuperconductivity but can also provide a mechanism for high Tc
superconductivity.
![Page 181: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/181.jpg)
Punch Line
◮ Long held belief: charge order competes and suppressessuperconductivity
◮ Electronic liquid crystal phases, not only can coexist withsuperconductivity but can also provide a mechanism for high Tc
superconductivity.
◮ Inhomogeneous phases: natural local pairing mechanism with purelyrepulsive interactions
![Page 182: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/182.jpg)
Punch Line
◮ Long held belief: charge order competes and suppressessuperconductivity
◮ Electronic liquid crystal phases, not only can coexist withsuperconductivity but can also provide a mechanism for high Tc
superconductivity.
◮ Inhomogeneous phases: natural local pairing mechanism with purelyrepulsive interactions
◮ This mechanism is not due to an infinitesimal instability
![Page 183: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/183.jpg)
Punch Line
◮ Long held belief: charge order competes and suppressessuperconductivity
◮ Electronic liquid crystal phases, not only can coexist withsuperconductivity but can also provide a mechanism for high Tc
superconductivity.
◮ Inhomogeneous phases: natural local pairing mechanism with purelyrepulsive interactions
◮ This mechanism is not due to an infinitesimal instability
◮ Underlying normal state is not a Fermi liquid and it does not havequasiparticles
![Page 184: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/184.jpg)
Punch Line
◮ Long held belief: charge order competes and suppressessuperconductivity
◮ Electronic liquid crystal phases, not only can coexist withsuperconductivity but can also provide a mechanism for high Tc
superconductivity.
◮ Inhomogeneous phases: natural local pairing mechanism with purelyrepulsive interactions
◮ This mechanism is not due to an infinitesimal instability
◮ Underlying normal state is not a Fermi liquid and it does not havequasiparticles
◮ Tc scales like a simple power of a coupling instead of an exponentialdependence
![Page 185: The Role of Charge Order in the Mechanism of High ...eduardo.physics.illinois.edu/homepage/ornl-charge-order...The Role of Charge Order in the Mechanism of High Temperature Superconductivity](https://reader036.vdocuments.mx/reader036/viewer/2022071215/6045567c38d04f684f09ce2b/html5/thumbnails/185.jpg)
Punch Line
◮ Long held belief: charge order competes and suppressessuperconductivity
◮ Electronic liquid crystal phases, not only can coexist withsuperconductivity but can also provide a mechanism for high Tc
superconductivity.
◮ Inhomogeneous phases: natural local pairing mechanism with purelyrepulsive interactions
◮ This mechanism is not due to an infinitesimal instability
◮ Underlying normal state is not a Fermi liquid and it does not havequasiparticles
◮ Tc scales like a simple power of a coupling instead of an exponentialdependence