Multifractal superconductivity - Institute for Nuclear Theory

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Nov 15, 2013 (3 years and 4 months ago)

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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