Can superconductivity emerge out
Can superconductivity emerge out
of a non Fermi liquid.
of a non Fermi liquid.
Andrey
Chubukov
University of Wisconsin
Washington University, January 29, 2003
Superconductivity
Ideal diamagnetism
Kamerling
Onnes, 1911
High Tc
superconductors
La2CuO4
Building blocks –
CuO2
layers
Phase diagram of the cuprates
Facts about high Tc
superconductors
•
Antiferromagnetism
of parent compounds(e.g, YBCO6 and La2CuO4)
•
dwave symmetry of the superconductiving
state
An exchange of near antiferromagnetic
spin fluctuations yields dwave pairing
(Scalapino, Pines,…)
(c.f. McMillan for phonons)
)
exp(
Tc
0
2
ξ
ξ
ξ
ξ
+
−
∝
−
Why there is still an interest in high
Tc
?
NonFermi liquid behavior in the normal state
Pseudogap
Fermi Liquid
•
Selfenergy
•
Resistivity
•
Optical conductivity
•
Specific heat
2
2
//
)
(
(
T
π
ω
+
∝
Σ
2)
D
in
ω
log
(ω
2
=
T
(T)
ρ
∝
2
)
(
ω
ω
σ
∝
T
C(T)
∝
Optimally doped Bi2212
2
'
'
not
,
)
(
ω
ω
ω
∝
Σ
Selfenergy vs
frequency and T
Linearity at large w
w/T scaling
Superconducting state
BCS theory
Photoemission intensity
normal state
k
F
I(
ω)
k
ω
0
∆
superconducting state
k
F
I(
ω)
k
ω
0
The superconducting gap vanishes at Tc
Photoemission intensity in high Tc
In a
The gap does not vanishes at Tc.
Pseudogap
(π,0)−(π,π)
8
5
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 al
Nature 382, 51 (1996)
1
7
0
K
8
5
K
1
0
K
3
0
0
K
τ(ω)
8
5
K
8
5
K
3
0
0
K
3
0
0
K
400
800
1200
1600
2000
0
1000
2000
1/τ(ω),
cm
1
400
800
1200
1600
G.Blumberg et al.
Science 278, 1427 (1997)
A.Puchkov et al
PRL 77, 3212 (1996)
1
0
K
1
0
K
STM
ARPES
IR:1/
Raman
YBa2Cu3O6.6
Tc= 59 K
6
5
K
3
0
0
K
0
400
800
1200
1600
2000
0
1000
2000
3000
1/τ(ω), [
cm]
1
0
400
800
1200
1600
2000
0
1000
2000
3000
4000
σ
1
(ω), (Ω
cm)
1
1/τ(ω) = ωp Re 1
_____
σ(ω)
2
0
100
200
300
T,
K
0
200
400
600
ρ
(T),
µΩ
cm
T*
1
0
K
3
0
0
K
1
0
K
6
5
K
Pseudogap: inplane scattering rate
cm 1
cm 1
Pseudogap
in the tunneling data for Bi2212
underdoped
overdoped
Strong coupling theories for the cuprates
Two different approaches depending on the point of departure
•
doping of a quantum antiferromagnet
(Mott insulator + interactions)
•
strong coupling spin fluctuation theory (Fermi liquid + interactions)
Another approach 
Marginal Fermi liquid phenomenology
The real issue is whether superconductivity, pseudogap
and
NonFermi liquid physics are all low energy phenomena
On one hand
eV
1
~
E
F
the upper scale for a Fermi liquid is
the effective interaction
eV
2

1
~
U
comparable
On the other hand
the superconducting gap
the pseudogap
temperature
nonFermi liquid behavior up to
F
E
0.08

0.04
~
∆
F
*
E
0.04

0.03
~
T
K
10
~
T
3
F
E
All these scales are at least order of magnitude smaller than
Let's see what the lowenergy approach gives us
•
is there a non FL behavior?
•
is there a superconductivity?
•
is there a pseudogap?
•
is there a secondary critical point?
Questions:
SPINFERMION MODEL
•
Describes the interaction between electrons and their own collective spin
degrees of freedom
Ingredients:
•
electrons near the Fermi surface
•
lowenergy collective spin excitations
•
a residual coupling between electrons and collective modes
Inputs:
•
Fermi velocity
•
spin correlation length
•
spinfermion
coupling
The model has two typical energy scales

effective interaction

internal energy scale
The ratio of the two determines the dimensionless coupling constant
2
sf
2
/4
ω
ω
λ
=
λ
D
−
3
λ
for arbitrary D
()
Perturbative
expansion in 2D holds in powers of
Problem with perturbation theory:
i.e., dimens
ionless coupling
diverges
at the quantum critical point.
ξ
λ
∝
Perturbation theory does not work in d=2 near the QCP
Back to the cuprates
Near optimal doping,
meV
20
~
sf
ω
NMR and neutrons
resonance neutron peak
meV
250

200
~
ω
sf
15

10
~
2,

1.5
~
ω
ω
λ
Even larger
λ
for underdoped
cuprates
For all relevant dopings, we are facing
the strong coupling problem, and
conventional weak coupling reasoning
is unapplicable
What to do when
∞
→
λ
?
Phonons
1
/v
v
1,
F
s
<
<
>>
λ
λ
Eliashberg
theory (solvable exactly)
Spin fluctuations
•
spin fluctuations have the same velocity as electrons
•
just one coupling
•
no Migdal
theorem
•
phonons are soft modes compared to electrons
•
two couplings
λ
and
F
s
/ v
v
λ
At strong coupling, spin fluctuations become diffusive
and soft compared to electrons
(0,π)
(π,0)
(π,π)
Q
h.s.
Fermi surface has hot spots 
points separated by
)
,
(
π
π
A spin fluctuation can decay into a
particlehole pair.
λ
and
Selfgenerated Eliashberg
theory

series in
)
(1
log
λ
+
analog of
F
s
/v
v
λ
Neglecting logs, we can solve the normal state exactly.
Eliashberg
theory
•
F
ermionic
and spin excitations vary at the same scale
sf
ω
10
8
6
4
2
0
ω
sf
ω/
Σ(ω) Im
(arb. units)
q2
χ ( ,ω) ∼ + ξ
2
1
q
Fermions:
Spin excitations:
q2
1
q
χ ( ,ω) ∼ + ω/ω
i
ω
G ( )~
ω
−1
=>
=>
ω
ω
ω
ω
2
1/2
=>
=>
2
FL
Fermions:
Spin excitations:
static
QC NFL
relaxational
0
sf
ω ∼ξ
Quantum Critical
NonFermi Liquid
Fermi Liquid
sf
Pairing problem
Spinmediated pairing yields attraction in dwave channel
(Scalapino, Pines…)
ω
ω
or
sf
Which of the two scales,
Temperature
doping
AF
AF
c
T
FL
AF
ins
T
ins
T
q.c. point
T
c
fluctuations
order parameter
quasiparticles only
pairing of Fermi liquid
pairing of nonFermi
liquid quasiparticles
determines the pairing instability?
sf
c
~
T
ω
Earlier reasoning
:
sf
ω
ω
<
•
only Fermi liquid regime is relevant,
•
effective coupling
•
pairing interaction decreases above
O(1)
)
/(1
eff
=
+
=
λ
λ
λ
sf
ω
)
exp(
Tc
0
2
ξ
ξ
ξ
ξ
+
−
∝
−
(c.f. McMillan for phonons)
Can nonFermi liquid fermions contribute to the pairing?
O(1)
)
/(1
eff
=
+
=
λ
λ
λ
in a Fermi liquid regime,
sf
ω
,
ω
eff
λ
above
remains constant up to
A novel, universal, non BCS pairing problem:
•
nonFermi liquid fermions
•
gapless spin collective mode
•
attaction
in a d
wave channel
Analytical and numerical analysis:
A linearized
gap equation has a solution at
ω
~
T
ins
0
0.5
1
1.5
2
inverse coupling
λ1
0
0.1
0.2
T/ω
−−
McMillan
Tins
∞
=
=
λ
ω
at
0.17
T
ins
The onset of the pairing instability
Do we have a true superconductor below
T
ins
?
0123
ω/ω
− −
0
0.5
units of ω
− −
Re∆(ω)
Im∆(ω)
1
2
3
4
λ=2, T=0
ReZ(ω)
ImZ
(ω)
4)
(0)/T
(2
T
~
0)
(T
ins
ins
≈
∆
=
∆
The gap
Phase fluctuations are irrelevant
(Fermi energy is the largest scale)
What is unusual?
Collective spin fluctuation modes
at energies below the gap
Low energy spin fluctuations in a superconductor
Normal state
sf
ω
overdamped
spin fluctuations at
Superconducting state
•
no lowenergy decay due to fermionic
gap
•
spin fluctuations become propagating
2
2
1
~
)
(
res
ω
ω
ω
χ
−
1
2
/
1
sf
res
~
)
(
~
ξ
ω
ω
ω
0)
(T
res
=
∆
≤
ω
Resonance peak in a dwave superconductor
Q:
For how long can coherent
superconductivity survive?
res
s
~
ω
ρ
T
(T)
res
s
−
∝
ω
ρ
A:
Up to
res
c
~
T
ω
•
At T=0, longitudinal superconducting stiffness
•A
t
T
>
0
,
•
The specific heat C(T) for a coherent state changes sign at
res
~
T
ω
Evidence:
Physics:
2
2
1
~
)
(
res
ω
ω
ω
χ
−
res
ω
attraction only up to
Conclusions
strong interaction between fermions and their own
lowenergy spin collective modes yields:
•
a pairing instability at Tins
~
•
nonFermi liquid, QC behavior in the normal state between
sf
ω
and
ω
ω
1/2
sf
)
(
ω
ω
that yields a very small gain in the condensation enegy
•
a true superconductivity at Tc
~
that scales with the resonance neutron frequency
Collaborators
•
Artem
Abanov
(UW/LANL)
•
Boris Altshuler
(Princeton)
•
Sasha
Finkelstein
(Weizmann)
•
R. Haslinger
(UW/LANL)
•
J. Schmalian
(Iowa)
•
E. Yuzbashuan
(Princeton)
0
200
400
600
8
00
Temperature, K
0
1
2
3
Resistivity, m
Ω
cm
PSEUDOGAP: aaxis resistivity
T*
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