eliashberg theory of spin-fluctuation pairing in cuprate...
TRANSCRIPT
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Andrey Chubukov
University of Wisconsin
Chernogolovka, October 9, 2010
Eliashberg theory of spin-fluctuation pairing in cuprate superconductors
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Dear Sima and Seva:
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Electron-doped
Hole-doped
High Tc cuprates
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Electron-doped
Hole-doped
• Parent compounds are antiferromagnetically ordered
• Overdoped, but still superconducting materials, are Fermi liquids
• Superconducting state has d-wave symmetry
Why spin fluctuations?
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Overdoped compounds are metals and Fermi liquids
Tl2Ba2CuO6+
Vignolle et al
Photoemission
Plate et al
Oscillations in resistance/magnetization
Areas are consistent with Luttinger count for electrons in a Fermi liquid (1+x)
X
G
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Campuzano et al
2 cos )( 0
0
0
0
0
Shen, Dessau et al 93, Campuzano et al, 96
kF
The superconducting gap has d-wave symmetry
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From the very early days of high-Tc
Scalapino et al, Monthoux,
Balatsky & Pines, Carbotte…
What if we replace phonons by antiferromagnetic spin fluctuations,
and perform BCS-type calculations?
(select spin channel)
=
p ,
'' p- ,
p ,
p- ,
'' p ,
p- ,
Analog of BCS
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+
Magnetic interaction is repulsive
q)-(k (q) E (q)
(q) q d (k) spin
22
-
tic)(paramagne
positive is q)-(kspin
No s-wave solution
Eqn. for asc gap
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+
Magnetic interaction is repulsive
q)-(k (q) E (q)
(q) q d (k) spin
22
-
) ,(at peaked is
q)-(k that Assume spin
Eqn. for asc gap
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+
Magnetic interaction is repulsive
q)-(k (q) E (q)
(q) q d (k) spin
22
-
gap d 2 2 yx
) ,(at peaked is
q)-(k that Assume spin
Eqn. for asc gap
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Objections:
• Cuprates near optimal doping (where Tc is the largest) areNOT weakly coupled Fermi liquids
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“Non-Fermi liquid” physics(most likely, Fermi liquid, but with small upper edge)
resistivity
T (T) ρ
22// )(
liquid, Fermi aIn
T
2T (T) ρ
liquid Fermi aIn
photoemission
)( ''
T. Valla et al
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Objections:
• Cuprates near optimal doping (where Tc is the largest) areNOT weakly coupled Fermi liquids
• There is pseudogap behavior over a wide range of temperatures
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Pseudogap
85 K
4.2 K
Bi2Sr2CaCu2O8
(Tc = 82 K)
500 0 -500 -1000 -1500 -2000 cm-1
-500 0 500 1000 1500 2000
Ch.Renner et al.PRL 80, 149 (1998)
dI/dV
H.Ding et alNature 382, 51 (1996)
170 K 85 K 10 K
300 K
STM
ARPES
IR:1/t
Raman
85 K
85 K
300 K
300 K
400 800 1200 1600 20000
1000
2000
tcm
-1
400 800 1200 1600
G.Blumberg et al.Science 278, 1427 (1997)
A.Puchkov et alPRL 77, 3212 (1996)
10 K
10 K
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Objections:
• Cuprates near optimal doping (where Tc is the largest) areNOT weakly coupled Fermi liquids
• Parent compounds of cuprates are not just antiferromagnets,they are also Mott insulators.
• There is pseudogap behavior over a wide range of temperatures
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Mott insulators
parent compounds(magnetic)
Resistivity
Ando et al
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Mott insulators
Onose et al,
Zimmers et al
The same Mott gap of 1.7 eV in hole-doped and el-doped materials
Optical conductivity
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What is more important, Mott physics or antiferromagnetism?(actually quite relevant to the issue of pairing mechanism)
• Doped Mott insulator is not a Fermi liquid (fermions are almost localized) – new pairing theory is required
• Doped Heisenberg antiferromagnet is a FermiLiquid with a small, pocketed Fermi surface –
one can study pairing using conventional theories
+ Small amount of holes
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What is more important, Mott physics or antiferromagnetism?(actually quite relevant to the issue of pairing mechanism)
• Doped Mott insulator is not a Fermi liquid (fermions are almost localized) – new pairing theory is required
• Doped Heisenberg antiferromagnet is a FermiLiquid with a small, pocketed Fermi surface –
one can study pairing using conventional theories
+Hole
pockets
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The crossover from small to large Fermi surface:
Hole
pockets Electron
pockets
A lot of fluctuations around these states, but Fermi liquid survives at T=0
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Physical Review B 81, 140505(R) (2010)
Suchitra E. Sebastian, N. Harrison,
M. M. Altarawneh, Ruixing Liang, D. A. Bonn,
W. N. Hardy, and G. G. Lonzarich
Evidence for small Fermi pockets
Original observation:
N. Doiron-Leyraud, C. Proust, D.
LeBoeuf, J. Levallois,
J.-B. Bonnemaison, R. Liang,
D. A. Bonn, W. N. Hardy,
and L. Taillefer,
Nature 447, 565 (2007)
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Evidence for small Fermi pockets
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L. Taillefer, cond-mat. 0901.2313
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d-wave
superconductor
An
tife
rrom
agnet
Incommensurate
antiferromagnetism
Evidence for antiferromagnetism
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d-wave
superconductor
An
tife
rrom
agnet
Incommensurate
antiferromagnetism
Evidence for antiferromagnetism
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Let’s take antiferromagnetism and change in Fermi
surface as the two ingredients of cuprate phase diagram
Quantum-criticalbehavior
Sachdev,…
Small Fermipockets
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Question: can we get high Tc superconductivity?
Quantum-criticalbehaviorSmall
Fermipockets
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Pairing mediated by strong antiferromagnetic spin fluctuations
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An exchange of antiferromagnetic spin fluctuations
yields d-wave pairing
Weak coupling: just replace phonons by spin fluctuations
)/-(1
sfc e ~ T K) 300- (200 meV 30-20 ~ sf 100K ~ T 1 c
It is possible to explain Tc ~ 100 K within weak/moderate coupling theory
Fermi liquid
Weak couplingBCS
Inconsistent
Need to go beyond BCS
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Eliashberg theory of spin fluctuation mediatedd-wave superconductivity
Search for “Eliashberg” yielded about 2500 papers only in PRB
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Model: fermions coupled to their collective
bosonic fluctuations in the spin channel
(spin-fermion model)
Ingredients: low-energy fermions ( ) , spin excitations ( )
and spin-fermion interaction
Parameters: one overall energy scale one dimensionless coupling
g1
Fv/ g
1Fv g
Strong coupling: when ,
The scale g is still assumed to be smaller than the fermionic bandwidth
(mass renormalization m*/m = 1 + )
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Why Eliashberg theory?
In bare theory, there is no difference between velocity of fermions and velocity of bosons – both are Fermi velocities, hence no Migdal theorem.
Once you dress up fermions AND bosons by self-energies, bosonsbecome Landau overdampred and hence slow compared to fermions
),k-(k v~ 0) (k, , ) ,(k FFF
, O(1) k)v()/ (k, ,1 )/ (k, F
Vertex corrections: g /g = O(1)
One needs large N to put theory under control [O(1) -> O(1/N)]
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Logics:
• compute normal state fermionic and bosonic propagators(with self-energies included)
• compare with experiments, extract and g
• use the renormalized propagators and Eliashbergtheory for the pairing problem, see what Tc andfeedbacks from the pairing on electrons we get
At this stage, no free parameters.
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Consider first doping region with no
Fermi surface reconstruction
Collective spin excitations can decay into particles
and holes and become Landau overdamped
Fermions acquire a finite damping due to interaction with Landau overdamped collective excitations
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Normal state
sf/ i - 1
1 Im ~ ) ,( Im
2
2
sf
g
64
9 g/
2-1.5 ~ meV 30-20 ev,7.1~ sf g
Spin fluctuations in the normal state form a gapless continuum
Fong et al
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1
2
Fermionic self-energy
Data: Kaminski et al
Nodal
MFL-like behavior
Anti-nodal
Fermi liquid is the upper boundary
of the Fermi liquid behaviorsf
sf
2
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Conductivity in YBCO7
theory
experiment(Basov et al)
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Quasiparticle dispersion from photoemission
Theory
Experimentpeak
hump
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Fermi liquid
-2
sf Quantum-critical
behavior
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upper boundary of a Fermi liquid
upper boundary of thestrong couping behavior
energy
g
The pairing problem
Two candidate scales
at vanishes
g ~ e ~ T -2)/ -(1
sfc
at finite
remains g, ~ Tc
2
g ~ sf
) (k, ) (k,
(if pairing is within a FL) (if pairing involves fermions outside of a FL)
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Pairing in non-Fermi liquid regime is a new phenomenon
2/1 )( 2/1
L )( )(/ 1
,)g/|(| 1
1
|-| ||
)( T
4 - )(
2/12/12/1
kk
Equation for the pairing vertex has non-BCS form
This is strong coupling limit of Eliashberg theory
This is NOT BCS – summing up logarithms gives no pairing at all
0Tat g
)(
4/1
0
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K) (200 at
g 0.025 T ins
Full solution (beyong logarithmic aproximation):the pairing instability exists at Tins ~ g
Fermi liquid pairing only, Tins ~ sf
T/g0.03
0.015
(same as the distance from a magnetic instability)
Abanov, A.C,Finkelstein
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This problem is actually quite generic
1/2
) (log 0
2
1/4
1/3
Antiferromagnetic QCP
Ferromagnetic QCP
2kF QCP
Pairing by near-gapless phonons
3D QCP, Color superconductivity
1 0
0.7
1 Z=1 pairing problem
Abanov, A.C., Finkelstein, hot spots
Haslinger et al, Millis et al, Bedel et al…2/3 problem: gauge field, nematic ….
Krotkov et al, electron-doped
Allen, Dynes, Carbotte, Marsiglio, Scalapino, Combescot, Maksimov, Bulaevskii, Rainer, Dolgov, Golubov, …
Son, Schmalian, A.C….
pairing in the presence of SDW Moon, Sachdev
fermions with Dirac cone dispersion Metzner et al
2
- 1 ,
)g/|(| 1
1
|-| ||
)( T )(
-1
0.18 T Dc
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Quantum-criticalbehavior
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Pairing in the presence of Fermi surface reconstruction
Tc decreases due to the reduction in spin-fermion vertex
MoonSachdev
Numbers match – their EF for POCKETS is 10 times smaller than g
Pairing by spin waves,
relevant scale is J ~100 meV
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Quantum-criticalbehavior
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Story is far from completion:
• Linear in resistivity is NOT explained
Quantum-criticalbehavior
Theory: resistivity in the critical regime is linear in T only above some T0
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• The nature of the pseudogap phase is not fully understood
It is natural to associate this phase with SDWprecursors (fluctuating pockets) , but
• There surely are superconducting fluctuations above Tc
• There are experimental evidences for potential discretesymmetry breaking in the pseudogap phase
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Neutron scattering: breaking of rotational symmetry
Could be pre-emptive ordering of Ising degree of freedomassociated with incommensurate magnetic order [(Q,) or (,Q)], but this remains to be seen.
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Conclusions
Recent experiments show that cuprates are more “conventional” than previously thought
• Coherent fermionic excitations are present at all dopings
• Long-range magnetic order extends up to optimal doping
This gives weight to the scenario that the pairing in the cuprates is mediated by near-critical spin fluctuations, and from this perspective is not that different from the pairing in heavy-fermion and organic superconductors
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Quantum-criticalbehavior
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• Artem Abanov (LANL/Texas)
• Sasha Finkelstein (Weizman/Texas A&M)
• Rob Haslinger (LANL)
• Dirk Morr (Illinois-Chicago)
• Joerg Schmalian (Iowa)
• Mike Norman (Argonne)
• Pavel Krotkov (Maryland)
• Ilya Eremin (Dresden)
• Karl Bennemann (Berlin)
• Oleg Tchernyshov (Baltimore)
• Boldizsar Janko (Notre Dame)
• David Pines (UC Davis)
• Philippe Monthoux (Edinburg)
• Matthias Eschrig (Karlsruhe)
• Tigran Sedrakyan (Maryland)
• A. Millis (Columbia)
• E. Abrahams (UCLA)
• S. Maiti (Wisconsin)
• D. Dhokarh (Wisconsin)
Collaborators
THANK YOU
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