Multifractal superconductivity
Vladimir Kravtsov, ICTP (Trieste)
Collaboration:
Michael Feigelman (Landau Institute)
Emilio Cuevas (University of Murcia)
Lev Ioffe (Rutgers)
Seattle, August 27, 2009
Superconductivity near localization
transition in 3d
int
ˆ
,
,
'
H
V
H
a
r
r
r
r
r
)
(
)
(
)
(
)
(
0
int
r
r
r
r
H
r
NO Coulomb interaction
3d lattice tight

binding model with diagonal disorder
Local tunable attraction
Relevant for cold atoms in disordered
optical lattices
i
)
(
i
f
2
/
W
2
/
W
Fermionic
atoms trapped
in an optical
lattice
Disorder is
produced by:
speckles
Other trapped
atoms
(impurities)
Cold atoms trapped in an optical lattice
New possibilities for superconductivity in
systems of cold atoms
1d localization is experimentally observed
[J.Billy et al., Nature
453
, 891, (2008); Roati
et al., Nature
453
, 895 (2008)]
2d and 3d localization is on the way
Tunable short

range interaction between
atoms: superconductive properties vs.
dimensionless attraction constant
No long

range Coulomb interaction
Strong controllable disorder,
realization of the Anderson model
Q: What does the strong
disorder do to
superconductivity?
A1: disorder gradually kills
superconductivity
A2: eventually disorder kills
superconductivity but before killing
it enhances it
Why superconductivity is possible when single

particle states are localized
3
/
1
0
1
)
(
T
T
R
Single

particle
conductivity:
only
states in the energy strip
~T near Fermi energy
contribute
R(T)
x
Superconductivity:
states in
the energy strip ~
D
near the
Fermi

energy contribute
)
0
(
)
(
~
)
(
D
T
R
T
R
R
c
R
(D)
)
(
c
T
R
x
)
(
c
T
R
x
Interaction sets in a new
scale
D
which stays
constant as
T

>0
Weak and strong disorder
V
W
/
L
disorder
Critical states
Localized
states
Extended
states
Anderson transition
F
~
Multifractality of critical and
off

critical states
)
1
(
2
1

)
(

n
d
r
n
i
n
L
r
W>W
c
W<E
c
Matrix elements
Interaction comes to play via matrix
elements
)
(
)
(
2
2
r
r
r
d
V
M
m
n
d
nm
Ideal metal and insulator
2
2
)
(
)
(
r
r
r
Vd
M
m
n
d
nm
Metal:
1
1
1
V
V
V
V
Small amplitude
100% overlap
Insulator:
1
1
1
V
V
d
d
d
d
x
x
x
x
Large amplitude
but rare overlap
Critical enhancement of correlations
d
d
E
E
/
1
2

'

Amplitude higher than in
a metal but almost full
overlap
States rather remote
(
\
E

E’<E
0
)
in energy
are strongly correlated
Simulations on 3D Anderson model
Ideal metal:
x
l
0
0
E
d
d
E
E
E
/
1
0
2
'
Multifractal metal:
x
l
0
x
W
W
W
c
c
F
m
n
m
n
m
n
m
n
m
n
d
E
E
E
E
E
E
E
E
r
r
r
d
V
E
E
C
,
2
2
,
)
'
(
)
(
)
'
(
)
(
)
(
)
(
)
'
(
(
)
c
W
D
E
~
1
3
0
0
0
(
)
1
0
d
x
x
Critical power law
persists
x
W=10
W=5
W=2
W
c
=16.5
3
/
1
0
~
c
F
W
Superconductivity in the vicinity of
the Anderson transition
Input:
statistics of multifractal states:
scaling (diagrammatics and sigma

model do not
work)
How does the superconducting transition
temperature depend on interaction
constant and disorder?
Mean

field approximation and the
Anderson theorem
)
;
'
,
(
)
'
(
'
)
(
T
r
r
K
r
dr
r
D
D
)
'
(
)
'
(
)
(
)
(
)
(
)
;
'
,
(
*
*
r
r
r
r
T
U
T
r
r
K
i
j
j
i
ij
ij
j
i
j
i
ij
E
E
T
E
T
E
T
)
2
/
tanh(
)
2
/
tanh(
)
(
ij
Wavefunctions drop out of the equation
Anderson theorem: Tc does not depend on
properties of wavefunctions
1/V
What to do at strong disorder?
D
(r) cannot be averaged
independently of K(r,r’)
Fock space instead of the real space
F
E
F
E
Superconducting phase
Normal phase
0
,
y
x
i
S
0
,
y
x
i
S
)
(
2
j
i
j
i
j
i
ij
z
i
i
i
eff
S
S
S
S
M
U
S
H
Single

particle
states,
strong
disorder
included
a
a
S
)
1
(
2
1
a
a
a
a
S
z
Why the Fock

space mean field is better than
the real

space one?
ij
j
j
j
j
i
M
E
T
E
D
D
)
2
/
tanh(
)
(
)
(
2
2
r
r
dr
M
j
i
ij
)
(
2
y
j
y
i
x
j
x
i
j
i
ij
z
i
i
i
eff
S
S
S
S
M
S
H
Infinite or large coordination number for
extended and weakly localized states
Weak fluctuations of M
ij
due to space integration
i
j
MF critical temperature close to
critical disorder
d
d
E
E
E
E
E
M
E
E
M
E
T
E
E
dE
E
/
1
0
2

'

)
'
(
)
'
(
'
)
2
/
'
tanh(
)
'
(
'
)
(
D
D
78
.
1
0
)
/
1
/(
1
0
~
~
2
E
E
T
d
d
c
>>
/
1
exp
D
At a small
parametrically large
e
nhancement of T
c
M.V.Feigelman, L.B.Ioffe, V.E.K. and E.Yuzbshyan,
Phys.Rev.Lett. v.98, 027001 (2007);
Thermodynamic phase fluctuations
dr
r
r
r
r
M
l
k
j
i
ijkl
)
(
)
(
)
(
)
(
Only possible if off

diagonal terms like
are taken into account
Ginzburg parameter
Gi~1
Expecting
1
~
/
c
c
T
T
D
Cannot kill the
parametrically
large
enhancement
of Tc
The phase diagram
Mobility
edge
Localized
states
Extended
states
T
c
(Disorder)
2
.
0
BCS
BUT…
r
e
2
Sweet life is only possible without
Coulomb interaction
Virial expansion method of calculating the
superconducting transition temperature
)
(
)
(
c
n
n
c
T
T
1
)
(
)
(
lim
1
c
n
c
n
n
T
T
Replacing by:
)
(
)
(
3
2
c
c
T
T
Operational definition of Tc for
numerical simulations
)
(
2
y
j
y
i
x
j
x
i
j
i
ij
z
i
i
i
eff
S
S
S
S
M
U
S
H
Tc at the Anderson transition: MF vs virial expansion
M.V.Feigelman, L.B.Ioffe, V.E.K. and E.Yuzbshyan,
Phys.Rev.Lett. v.98, 027001 (2007);
d
d
E
c
T
crit
/
1
,
2
/
1
0

'

E
E
Virial
expansion
MF result
2d Analogue: the Mayekawa

Fukuyama

Finkelstein effect
The diffuson diagrams
The cooperon diagrams
Superconducting transition temperature
Virial expansion on the 3d Anderson model
metal
insulator
(disorder)
Anderson
localization
transition
metal
insulator
Conclusion:
Enhancement of Tc by disorder
Maximum of Tc in the insulator
Direct superconductor to
insulator transition
BUT
Fragile superconductivity:
Small fraction of superconducting phase
Critical current decreasing with disorder
Two

eigenfunction correlation in 3D
Anderson model (insulator)
x
1
ln
)
(
d
NC
Ideal insulator
limit only in
one

dimensions
d
d
E
E
/
1
2

'

critical, multifractal
physics
Mott’s resonance
physics
x
Superconductor

Insulator transition: percolation without
granulation
x
)
(
crit
T
c
x
)
(
crit
T
c
c
T
x
c
T
x
SC
INS
Only states in the
strip ~T
c
near the
Fermi level take part
in superconductivity
Coordination
number K>>1
Coordination
number K=0
First order transition?
c
c
c
T
U
T
M
T
K
~
)
(
)
(
T
T
T
T
f
T
T
T
T
E
T
f
T
KMU
d
d
)
/
(
)
/
(
~
~
)
/
(
)
/
(
)
/
(
~
/
0
0
0
2
x
x
x
x
x
Tc
f(x)

>0 at x<<1
Conclusion
Fraclal texture of eigenfunctions persists in metal and
insulator (multifractal metal and insulator).
Critical power

law enhancement of eigenfunction
correlations persists in a multifractal metal and
insulator.
Enhancement of superconducting transition
temperature
due to critical wavefunction correlations.
disorder
Anderson
localization
transition
BCS
limit
SC
SC
M
IN
SC or IN
T
c
c
c
T
U
T
M
T
K
~
)
(
)
(
T
T
d
d
T
T
T
T
E
T
KMU
/
0
0
0
2
)
/
(
~
)
/
)(
/
(
~
x
x
x
KMU
Corrections due to off

diagonal
terms
ijkl
kl
kl
kl
ij
M
T
U
)
(
D
D
D
D
m
k
k
llkm
iikm
km
il
l
l
l
i
M
M
U
M
T
U
,
)
(
Average value of
the correction term
increases T
c
Average
correction is
small when
1
2
2
4
2
3
d
d
d
d
Melting of phase by disorder
)
(
)
(
)
(
r
r
r
j
i
ij
ij
ij
D
D
In the diagonal
approximation
)
(
)
2
/
tanh(
)
(
2
r
T
E
r
i
i
i
i
D
D
D
D
m
k
k
llkm
iikm
km
il
l
l
l
i
M
M
U
M
T
U
,
)
(
The sign correlation <
D
(r)
D
(r’)>
is perfect : solutions
D
i>0
do not lead to a global phase destruction
stochastic term
destroys phase
correlation
Beyond the
diagonal
approximation:
How large is the stochastic term?
m
k
k
llkm
iikm
km
il
M
M
U
Q
,
Stochastic term:
2
2
il
il
Q
Q
d
2
>
d
/
2
d
2
>d/2 weak oscillations
d
2
< d/2 strong oscillations
but still too small to support
the glassy solution
2
2
/
2
1
2
~
~
2
N
Q
N
Q
il
d
d
il
d
2
<d/2
More research is needed
Conclusions
Mean

field theory beyond the Anderson theorem:
going into the Fock space
Diagonal and off

diagonal matrix elements
Diagonal approximation: enhancement of T
c
by
disorder.
Enhancement is due to sparse single

particle
wavefunctions and their strong correlation for
different energies
Off

diagonal matrix elements and stochastic term in
the MF equation
The problem of “cold melting” of phase for d
2
<d/2
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