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The Role of Charge Order in the Mechanism of HighTemperature Superconductivity
Eduardo FradkinDepartment of Physics
University of Illinois at Urbana-Champaign
Steven Kivelson, UCLA/Stanford
Enrico Arrigoni, TU Graz, Austria
Victor J. Emery∗, Brookhaven
Condensed Matter Physics Seminar
National High Magnetic Field Laboratory, Florida State University, Tallahassee, April 27, 2007
Steven Kivelson and Eduardo Fradkin, How optimal inhomogeneity produces high temperature
superconductivity; arXiv:cond-mat/0507459.
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References• Steven A. Kivelson and Eduardo Fradkin, How optimal inhomogeneity produces high
temperature superconductivity; to appear as a chapter inTreatise of High Temperature
Superconductivity, by J. Robert Schrieffer and James Brooks, in press (Springer-Verlag, 2007);
arXiv:cond-mat/0507459.
• Enrico Arrigoni, Eduardo Fradkin and Steven A. Kivelson, Mechanism of High TemperatureSuperconductivity in a striped Hubbard Model, Phys. Rev. B69, 214519 (2004);arXiv:cond-mat/0309572
• Steven A. Kivelson, Eduardo Fradkin, Vadim Oganesyan, Ian Bindloss, 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, Electronic Liquid Crystal Phases ofa Doped Mott Insulator, Nature393, 550 (1998); arXiv:cond-mat/9707327.
• Victor J. Emery, Eduardo Fradkin, Steven A. Kivelson and TomC. Lubensky, Quantum Theoryof the Smectic Metal State in Stripe Phases, Phys. Rev. Lett.85, 2160 (2000);arXiv:cond-mat/0001077
• Mats Granath, Vadim Oganesyan , Steven A. Kivelson, EduardoFradkin and Victor J. Emery,Nodal quasi-particles and coexisting orders in striped superconductors, Phys. Rev. Lett.87,167011 (2001); arXiv:cond-mat/0010350.
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Outline• Inhomogeneous Phases areinherentto Doped Mott Insulators
• Charge Order and Superconductivity: friend or foe?
• High Tc Superconductivityin a Striped Hubbard Model
• Isolated doped Hubbard Ladders as theprototype spin-gap systems
• EstimatingTc: How high is high?
• Conclusions and Open Questions
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What do experiments tell us
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P. Abbamonteet al, Nature Physics1, 155 (2005)
Stripe charge orderin underdoped high temperature superconductors (La2−xSrxCuO4, non-SC
La2−xBaxCuO4and YBa2Cu3O6+y) (Tranquada (neutrons), Mook (neutrons), Ando (transport),
Abbamonte (resonant X-ray scattering))
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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. Kivelsonet al, Rev. Mod. Phys.75, 1201 (2003)
Coexistence offluctuatingstripe charge order and superconductivityin LSCO and YBCO (Mook,
Tranquada)
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Similarity of high energy neutron spectra in La2−xSrxCuO4, non-SC La2−xBaxCuO4,
YBa2Cu3O6+yand the ladder material Sr14Cu24O41 (Tranquada, Mook, Keimer, Buyers)
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T. Valla et al, Science314, 1914 (2006).
Evidence ofthe existence of an optimal degree of inhomogeneity in La2−xBaxCuO4 (ARPES)
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J. Hoffmanet al, Science295, 466 (2002)
Induced charge order in the SC phase in vortex halos: neutrons in La2−xSrxCuO4(B. Lake), STM in
Bi2Sr2CaCu2O8+δ(Davis)
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C. Howaldet al, PNAS100, 9705 (2003)
STM Experiments:short range stripe order(on scaleslong compared toξ0), possible broken
rotational symmetry (Bi2Sr2CaCu2O8+δand Ca2−xNaxCuO2Cl2) (Kapitulnik, Davis, Yazdani)
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Kohsakaet al, Science315, 1380 (2007)Short range stripe order in Dy-Bi2Sr2CaCu2O8+δ
STM R maps at 150 mV
R(~r, 150mV ) = I(~r, +150mV )/I(~r,−150mV ).
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C. Panagopouloset al, PRL96, 040702 (2006); J. Bonettiet al, PRL93, 037002 (2004)
Transport experiments give evidence forcharge domain switching in YBa2Cu3O6+ynanowires(Van
Harlingen and Weissmann) andhysteretic effectsin the “normal state” of
La2−xSrxCuO4(Panagopoulos)
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Electron Liquid Crystal PhasesS. Kivelson, E. Fradkin, V. Emery, Nature393, 550 (1998)
Doping a Mott insulator: inhomogeneous phases arise due to the competition
betweenphase separationandstrong correlations
• Crystal Phases: break all continuous translation symmetries and rotations
• Smectic (Stripe)phases: break one translation symmetry and rotations
• NematicandHexaticPhases: are uniform and anisotropic
• Uniform fluids: break no spatial symmetries
High Tc Superconductors: Lattice effects⇒ breaking of point group
symmetries
If lattice effects are weak(highT ) ⇒ continuous symmetries essentially
recovered
2DEG in GaAs heterostructures⇒ continuous symmetries
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Electronic nematic state in the 2DEG in high magnetic fields
J. Eisensteinet al, PRB82, 394 (1999)
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Electronic nematic phase in Sr3Ru2O7
R. Borzi, S. Grigera, A. P. Mackenzieet alScience,315, 214 (2007).
B. Keimer finds a similarnematic state in YBa2Cu3O6+yaty = 0.4 for T . 150K!.
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Electronic Liquid Crystal Phases in HighTcSuperconductors
• Liquid: Isotropic, breaks no spacial symmetries; either a conductor or a
superconductor
• Nematic: Lattice effects reduce the symmetry to a rotations byπ/2
(“Ising”); translation and reflection symmetries are unbroken; it is an
anisotropic liquid with a preferred axis
• Smectic: breaks translation symmetry only in one direction but liquid-like
on the other; Stripe phase; (infinite) anisotropy of conductivity tensor
• Crystal(s): electron solids (“CDW”); insulating states.
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Tem
pera
ture
Nematic
Isotropic (Disordered)
Superconducting
C C12C3
hω
Crystal
Smectic
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Nematic
IsotropicSmectic
Crystal
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Soft Quantum Matter
or
Quantum Soft Matter
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Stripe Phases and the Mechanism of highTcsuperconductivity in Strongly Correlated Systems
Since the discovery of highTc superconductivity it has been clear that
• High Tc Superconductors are never normal metalsanddon’t have well
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
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ProblemBCS is so successful in conventional metalsthat the termmechanismnaturally
evokes the idea of aweak coupling instabilitywith (write here your favorite
boson) mediating an attractive interaction betweenwell defined quasiparticles.
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Superconductivity in a Doped Mott Insulatoror
How To Get Pairing from Repulsive Interactions
• Universal assumption: 2D Hubbard-like models should contain the essential
physics
• “RVB” mechanism:
– Mott insulator: spins are bound in singlet valence bonds; itis a strongly
correlated spin liquid, essentially apre-paired insulating state
– spin-charge separation in the doped state leads to highTc
superconductivity
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Problems
• there is no real evidence that the simple 2D Hubbard model favors
superconductivity(let alone highTc superconductivity)
• all evidence indicates that if anything it wants to be an insulator and to phase
separate(finite size diagonalizations, various Monte Carlo simulations)
• strong tendency for the ground states to be inhomogeneous and possibly
anisotropic
• no evidence (yet) for a spin liquid in 2D Hubbard-type models
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Why an Inhomogeneous State is Good for highTcsuperconductivity
• An “inhomogeneity induced pairing” mechanism of highTc superconductivity in
which the pairing of electrons originates directly from strong repulsive interactions.
• Repulsive interactions lead to local superconductivity on‘mesoscale structures’
• The strength of this pairing tendency decreases as the size of the structures increases
above an optimal size
• The physics responsible for the pairing within a structure⇒ Coulomb frustrated
phase separation⇒ mesoscale electronic structures
• 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 toT = 0
– The highest possibleTc is obtained with an intermediate degree of
inhomogeneity
– The optimalTc 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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Stripe Hubbard ModelA Cartoon of the Strongly Correlated Stripe Phase
H = −X
<~r,~r′>,σ
t~r,~r′
hc†~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
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Physics of the 2-leg ladder
t
t′
U
V
E
p
EF
pF1−pF1
pF2−pF2
• ForU = V = 0 there are two bands of states
• The bands have different Fermi wave vectors,pF1 6= pF2
• The only allowed scattering processes involve anevennumber of electrons
(momentum conservation)
• The coupling of CDW fluctuations withQ1 = 2pF1 6= Q2 = 2pF2 is suppressed
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Why is there a Spin Gap• Scattering of electron pairs with zero center of mass momentum from one system to
the other is peturbatively relevant
• The electrons can gain zero-point energy by delocalizing between the two 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-gap phase.
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What is it known about the 2-leg ladder• At x = 0 there is aunique fully gapped ground state(“C0S0”); for U ≫ t,
∆s ∼ J/2
• For0 < x < xc ∼ 0.3, Luther-Emery liquid: no charge gap and large spin gap
(“C1S0”); spin gap∆s ↓ asx ↑, and∆s → 0 asx → xc
• Effective Hamiltonian for the charge degrees of freedom
H =
Zdy
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, andµ depends ont′/t andU/t
• . . . represent cosine potentials: Mott gap∆M atx = 0
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• Excitations forx → 0 are spinless charge2e fermionic solitons
• K → 2 asx → 0; K ∼ 1 for x ∼ 0.1, andK ∼ 1/2 for x ∼ xc
• ladder superconducting suceptibility:χSC ∼ ∆s/T 2−K−1
• ladder CDW suceptibility:χCDW ∼ ∆s/T 2−K
• χCDW(T ) → ∞ andχSC(T ) → ∞ for 0 < x < xc
• Forx . 0.1, χSC ≫ χCDW!
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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 inδt:
– marginal(and small) forward scattering inter-ladder interactions
– Relevant Perturbations: Inter-ladder Josephson couplingsandInter-ladder CDW
couplings
H ′ = −X
J
Zdy
hJ cos
“√2π∆θJ)
”+ V cos
“∆PJy +
√2π∆φJ)
”i
J : ladder index;PJ = 2πxJ , ∆φJ = φJ+1 − φJ , etc.
– J andV are effective couplings which must be computed from microscopics
– Estimate:J ≈ V ∝ (δt)2/J
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Period 2 works forx ≪ 1
• If all the ladders are equivalent, a period 2 stripe ordered or column state
• For an isolated ladderTc = 0
• J 6= 0 andV 6= 0, TC > 0
• Forx . 0.1 CDW couplings are irrelevant(1 < K < 2): Inter-ladder
Josephson coupling leads to a superconducting state in a restricted range of
smallx with rather lowTc
2JχSC(Tc) = 1
• Tc ∝ δtx
• For largerx, K < 1 andχCDW is more strongly divergentthanχSC
• CDW couplings become more relevant⇒ Insulating, incommensurate
CDW state with ordering wave numberP = 2πx.
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Why Period 4 works• Consider an alternating array of A and B type ladders (with different
electron affinities) in the LE regime
• SCTc:
(2J )2χA
SC(Tc)χB
SC(Tc) = 1
• CDW Tc:
(2V)2χA
CDW(P, Tc)χB
CDW(P, Tc) = 1
• 2D CDW order is greatly suppressed due to the mismatch between ordering
vectors,PA andPB, on neighboring ladders
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• For inequivalent ladders SC beats CDW if
2 > K−1
A + K−1
B − KA; 2 > K−1
A + K−1
B − KB
•Tc ∼ ∆s
„JfW
«α
; α =2KAKB
[4KAKB − KA − KB]
• J ∼ δt2/J andfW ∼ J ; Tc is (power law) small for smallJ ! ( α ∼ 1).
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How reliable are these estimates?• This is amean-field estimatefor Tc and it is anupper boundto the actualTc.
• Tc should be suppressed by phase fluctuationsby up to a factor of 2.
• Indeed,perturbative RG studiesfor smallJ yield thesame power law
dependence. This result isasymptotically exactfor J << W̃ .
• SinceTc is a smooth function ofδt/J , it is reasonable toextrapolate for
δt ∼ J .
• ⇒ Tmaxc ∝ ∆s ⇒highTc.
• This is in contrast to theexponentially smallTc as obtained in aBCS-like
mechanism.
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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 dopingx
• xc(2) andxc(4) indicates the SC-CDW quantum phase transition for the period 2
and period 4 cases
• Forx & xc the isolated ladders do not have a spin gap; in this regime thephysics is
different involving low-energy spin fluctuations
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Open Questions• a period 2 modulation can produce superconductivity with arelatively low
Tc in a restricted doping range
• a period 4 modulation produceshigherTc’s on a broader range of doping
• Tc is only power-law small, with α ∼ 1
• no exponential suppression ofTc ⇒ “high Tc”
• This model is cartoon of the symmetry breaking of stripe (smectic) state
• It has alarge spin gapand it does not have low-energy spin fluctuations
• the order-parameter isd-wave like: it changes sign underπ/2 rotations
• It does not havenodal fermionic excitations
• Nodal fermions may appear upon a (Lifshitz) transition at largerx
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Punch Line• Long held belief: charge order competes and suppresses superconductivity
• Electronic liquid crystal phases, not onlycan coexist with superconductivity
but can alsoprovide a mechanism for highTc superconductivity.
• Inhomogeneous phases:natural local pairing mechanismwith purely
repulsive interactions
• This mechanismis notdue to aninfinitesimal instability
• Underlying normal stateis not a Fermi liquidandit does not have
quasiparticles
• Tc scales like a simplepowerof a coupling instead of an exponential
dependence
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