Can superconductivity emerge out Can superconductivity emerge out of a non Fermi liquid. of a non Fermi liquid.

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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)

d-wave symmetry of the superconductiving
state
An exchange of near antiferromagnetic
spin fluctuations yields d-wave 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
?

Non-Fermi liquid behavior in the normal state

Pseudogap
Fermi Liquid

Self-energy

Resistivity

Optical conductivity

Specific heat
2
2
//
)
(
(
T
π
ω
+

Σ
2)
D
in

ω

log


2
=
T


(T)

ρ

-2

)
(

ω
ω
σ

T


C(T)

Optimally doped Bi2212
2
'
'
not

,

)
(

ω
ω
ω

Σ
Self-energy 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: in-plane 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
Non-Fermi 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
non-Fermi 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 low-energy approach gives us

is there a non FL behavior?

is there a superconductivity?

is there a pseudogap?

is there a secondary critical point?
Questions:
SPIN-FERMION MODEL

Describes the interaction between electrons and their own collective spin
degrees of freedom
Ingredients:

electrons near the Fermi surface

low-energy collective spin excitations

a residual coupling between electrons and collective modes
Inputs:

Fermi velocity

spin correlation length

spin-fermion
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
particle-hole pair.
λ
and
Self-generated 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
Non-Fermi Liquid
Fermi Liquid
sf
Pairing problem
Spin-mediated pairing yields attraction in d-wave 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 non-Fermi
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 non-Fermi 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:

non-Fermi 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
λ1
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 low-energy 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 d-wave 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
low-energy spin collective modes yields:

a pairing instability at Tins
~

non-Fermi 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

cm
PSEUDOGAP: a-axis resistivity
T*