the role of charge order in the mechanism of high...

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

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)

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.

Outline

◮ Electronic liquid crystal phases are unavoidable in doped Mottinsulators

Outline

◮ Experimental evidence for electronic liquid crystal phases in the highTc superconductors, Sr3Ru2O7, and quantum Hall systems

Outline

◮ Charge Order and Superconductivity: friend or foe?

Outline

◮ If inhomogeneous phases are part of the mechanism of hightemperature superconductivity, is there an optimal degree ofinhomogeneity?

Outline

◮ High Tc Superconductivity in a Striped Hubbard Model

Outline

◮ Isolated doped Hubbard Ladders as the prototype spin-gap systems

Outline

◮ Estimating Tc : How high is high?

Outline

◮ Conclusions and Open Questions

Outline

◮ Conclusions and Open Questions

Phase Diagram of the High Tc Superconductors

“pseudogap”

Superconductor

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

◮ 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

Electronic Liquid Crystal Phases in High Tc

Superconductors

◮ Liquid: Isotropic, breaks no spacial symmetries; either a conductoror a superconductor

Superconductors

◮ Nematic: Lattice effects reduce the symmetry to a rotations by π

(“Ising”); translation and reflection symmetries are unbroken; it is ananisotropic liquid with a preferred axis

Superconductors

◮ Smectic: breaks translation symmetry only in one direction butliquid-like on the other; Stripe phase; (infinite) anisotropy ofconductivity tensor

Superconductors

◮ Crystal(s): electron solids (“CDW”); insulating states

Superconductors

◮ Crystal(s): electron solids (“CDW”); insulating states

Electronic Liquid Crystal Phases in HTSC, cont’d

Soft Quantum Condensed Matter!

Schematic Phase Diagram of Doped Mott Insulators

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

◮ spin stripe ⇒ magnetic Bragg peaks at

~k = ~Qs = (π, π) ± 1

~k = ±~Qch = ± 2π

~k = ~Qs = (π, π) ± 1

◮ Order Parameter: 〈n~Qch

〉, Fourier component of the electron density

at ~Qch

~k = ±~Qch = ± 2π

~k = ~Qs = (π, π) ± 1

at ~Qch

◮ Smectic (stripe) order couples linearly to disorder ⇒ macroscopicglassy behavior

~k = ±~Qch = ± 2π

~k = ~Qs = (π, π) ± 1

at ~Qch

◮ Smectic (stripe) order couples linearly to disorder ⇒ macroscopicglassy behavior

Order Parameters for the Nematic State

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

◮ 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

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)

Fluctuating stripe charge order and superconductivity

I 7K (

-0.2 0.0 0.2

I 7K-I

-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)

-0.2 0 0.2

T = 1.5 KT = 50 K

Intensity (arb. units)100

-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

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

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)

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)

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

La2−xBaxCuO4

T. Valla et al, Science 314, 1914 (2006) (ARPES)

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)

Dynamical decoupling of the SC layers in La2−xBaxCuO4

��

◮ 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!

��

◮ 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)

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)

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)

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)

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

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

Fluctuating Stripes as Nematic Order in underdopedYBa2Cu3O6+y

◮ Fluctuating stripe order detected in inelastic neutron scattering,Mook et al (2000)

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)

Stripe Phases and the Mechanism of high Tc

superconductivity in Strongly Correlated Systems

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

◮ the “parent compounds” are strongly correlated Mott insulators◮ repulsive interactions dominate

◮ the “parent compounds” are strongly correlated Mott insulators◮ repulsive interactions dominate◮ the quasiparticles are an ‘emergent’ low-energy property of the

superconducting state

superconducting state◮ whatever “the mechanism” is has to account for these facts

Problem

◮ BCS is so successful in conventional metals

Problem

◮ BCS is so successful in conventional metals◮ that the term mechanism naturally evokes the idea of a weak

coupling instability

Problem

coupling instability◮ with (write here your favorite boson) mediating an attractive

interaction between well defined quasiparticles

Problem

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.

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:

◮ “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

◮ spin-charge separation in the doped state leads to high Tc

superconductivity

Problems

◮ there is no real evidence that the simple 2D Hubbard model favorssuperconductivity (let alone high Tc superconductivity)

Problems

◮ all evidence indicates that if anything it wants to be an insulator andto phase separate (finite size diagonalizations, various Monte Carlosimulations)

Problems

◮ strong tendency for the ground states to be inhomogeneous andpossibly anisotropic

Problems

◮ no evidence (yet) for a spin liquid in 2D Hubbard-type models

Problems

◮ no evidence (yet) for a spin liquid in 2D Hubbard-type models

Why an Inhomogeneous State is Good for high Tc SC

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

◮ 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

is suppressed by phase fluctuations, often to T = 0◮ The highest possible Tc is obtained with an intermediate degree of

inhomogeneity

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’

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’

Striped Hubbard Model: A Cartoon of the StronglyCorrelated Stripe Phase

H = −

<~r ,~r ′>,σ

t~r ,~r ′

~r,σc~r ′,σ + h.c.

~r ,σ

ǫ~r c†

~r ,σc~r ,σ +

~r,σc†

~r ,−σc~r ,−σc~r ,σ

t′t′ δtδtδt

εε −ε−ε

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.

Physics of the 2-leg ladder

pF1−pF1

pF2−pF2

pF1−pF1

pF2−pF2

◮ For U = V = 0 there are two bands of states

pF1−pF1

pF2−pF2

◮ The bands have different Fermi wave vectors, pF1 6= pF2

pF1−pF1

pF2−pF2

◮ The only allowed scattering processes involve an even number ofelectrons (momentum conservation)

pF1−pF1

pF2−pF2

◮ The coupling of CDW fluctuations with Q1 = 2pF1 6= Q2 = 2pF2 issuppressed

pF1−pF1

pF2−pF2

◮ The coupling of CDW fluctuations with Q1 = 2pF1 6= Q2 = 2pF2 issuppressed

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

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

[K (∂yθ)2 +

K(∂xφ)2

]+ . . .

φ: CDW phase field; θ: SC phase field; [φ(y ′), ∂yθ(y)] = iδ(y − y ′)

[K (∂yθ)2 +

K(∂xφ)2

]+ . . .

◮ x-dependence of ∆s , K , vc , and µ depends on t ′/t and U/t

[K (∂yθ)2 +

K(∂xφ)2

]+ . . .

◮ . . . represent cosine potentials: Mott gap ∆M at x = 0

[K (∂yθ)2 +

K(∂xφ)2

]+ . . .

◮ K → 2 as x → 0; K ∼ 1 for x ∼ 0.1, and K ∼ 1/2 for x ∼ xc

[K (∂yθ)2 +

K(∂xφ)2

]+ . . .

◮ χSC ∼ ∆s/T 2−K−1

χCDW ∼ ∆s/T 2−K

[K (∂yθ)2 +

K(∂xφ)2

]+ . . .

◮ χSC ∼ ∆s/T 2−K−1

◮ χCDW(T ) → ∞ and χSC(T ) → ∞ for 0 < x < xc

[K (∂yθ)2 +

K(∂xφ)2

]+ . . .

◮ χSC ∼ ∆s/T 2−K−1

◮ For x . 0.1, χSC ≫ χCDW!

[K (∂yθ)2 +

K(∂xφ)2

]+ . . .

◮ χSC ∼ ∆s/T 2−K−1

◮ For x . 0.1, χSC ≫ χCDW!

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:

◮ Second order processes in δt:◮ marginal (and small) forward scattering inter-ladder interactions

◮ Second order processes in δt:◮ marginal (and small) forward scattering inter-ladder interactions◮ Josephson couplings, possibly relevant

◮ 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 cos

(√2π∆θJ)

)+ V cos

(∆PJy +

√2π∆φJ)

J: ladder index; PJ = 2πxJ , ∆φJ = φJ+1 − φJ , etc.

H ′ = −∑

[J cos

(√2π∆θJ)

)+ V cos

(∆PJy +

√2π∆φJ)

◮ J and V are effective couplings which must be computed frommicroscopics

H ′ = −∑

[J cos

(√2π∆θJ)

)+ V cos

(∆PJy +

√2π∆φJ)

◮ Estimate: J ≈ V ∝ (δt)2/J

H ′ = −∑

[J cos

(√2π∆θJ)

)+ V cos

(∆PJy +

√2π∆φJ)

◮ Estimate: J ≈ V ∝ (δt)2/J

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

◮ J 6= 0 and V 6= 0, TC > 0

2JχSC(Tc) = 1

◮ Tc ∝ δt x

◮ J 6= 0 and V 6= 0, TC > 0

2JχSC(Tc) = 1

◮ Tc ∝ δt x

◮ For larger x , K < 1 and χCDW is more strongly divergent than χSC

◮ J 6= 0 and V 6= 0, TC > 0

2JχSC(Tc) = 1

◮ Tc ∝ δt x

◮ CDW couplings become more relevant ⇒ Insulating,incommensurate CDW state with ordering wave number P = 2πx .

◮ J 6= 0 and V 6= 0, TC > 0

2JχSC(Tc) = 1

◮ Tc ∝ δt x

◮ CDW couplings become more relevant ⇒ Insulating,incommensurate CDW state with ordering wave number P = 2πx .

Why Period 4 works even better!

Why Period 4 works even better!◮ Consider an alternating array of inequivalent A and B type ladders in

the LE regime

the LE regime◮ SC Tc :

(2J )2χASC(Tc)χ

BSC(Tc) = 1

(2J )2χASC(Tc)χ

BSC(Tc) = 1

◮ CDW Tc :(2V)2χA

CDW(P , Tc )χBCDW(P , Tc) = 1

(2J )2χASC(Tc)χ

BSC(Tc) = 1

◮ 2D CDW order is greatly suppressed due to the mismatch betweenordering vectors, PA and PB , on neighboring ladders

(2J )2χASC(Tc)χ

BSC(Tc) = 1

◮ For inequivalent ladders SC beats CDW if

(2J )2χASC(Tc)χ

BSC(Tc) = 1

2 > K−1A + K−1

B − KA; 2 > K−1A + K−1

B − KB

(2J )2χASC(Tc)χ

BSC(Tc) = 1

2 > K−1A + K−1

B − KA; 2 > K−1A + K−1

B − KB

Tc ∼ ∆s

; α =2KAKB

[4KAKB − KA − KB ]

(2J )2χASC(Tc)χ

BSC(Tc) = 1

2 > K−1A + K−1

B − KA; 2 > K−1A + K−1

B − KB

Tc ∼ ∆s

; α =2KAKB

[4KAKB − KA − KB ]

◮ J ∼ δt2/J and W ∼ J;Tc is (power law) small for small J ! (α ∼ 1).

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.

Schematic Phase Diagram for Period 2 and Period 4

∆s(x)

SC CDW

xxcxc(2) xc(4)

∆s(x)

SC CDW

xxcxc(2) xc(4)

◮ The broken line is the spin gap ∆s(x) as a function of doping x

∆s(x)

SC CDW

xxcxc(2) xc(4)

◮ xc(2) and xc(4) indicates the SC-CDW quantum phase transition forperiod 2 and period 4

∆s(x)

SC CDW

xxcxc(2) xc(4)

◮ 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

Open Questions

◮ a period 2 modulation can produce superconductivity with arelatively low Tc in a restricted doping range

Open Questions

◮ a period 4 modulation produces higher Tc ’s on a broader range ofdoping

Open Questions

◮ Tc is only power-law small, with α ∼ 1

Open Questions

◮ no exponential suppression of Tc ⇒ “high Tc ”

Open Questions

◮ This model is cartoon of the symmetry breaking of stripe (smectic)state

Open Questions

◮ It has a large spin gap and it does not have low-energy spinfluctuations

Open Questions

◮ the order-parameter is d-wave like: it changes sign under π/2rotations

Open Questions

◮ It does not have nodal fermionic excitations

Open Questions

◮ It does not have nodal fermionic excitations

◮ Nodal fermions may appear upon a (Lifshitz) transition at larger x

Punch Line

◮ Long held belief: charge order competes and suppressessuperconductivity

Punch Line

◮ Electronic liquid crystal phases, not only can coexist withsuperconductivity but can also provide a mechanism for high Tc

superconductivity.

Punch Line

superconductivity.

◮ Inhomogeneous phases: natural local pairing mechanism with purelyrepulsive interactions

Punch Line

superconductivity.

◮ This mechanism is not due to an infinitesimal instability

Punch Line

superconductivity.

◮ Underlying normal state is not a Fermi liquid and it does not havequasiparticles

Punch Line

superconductivity.

◮ Tc scales like a simple power of a coupling instead of an exponentialdependence

Punch Line

superconductivity.

◮ Tc scales like a simple power of a coupling instead of an exponentialdependence

the role of charge order in the mechanism of high...

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