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

intˆ,,' HVH arrrrr

)()()()(0

int rrrrHr

NO Coulomb interaction

3d lattice tight-binding model with diagonal disorder

Local tunable attraction

Relevant for cold atoms in disordered optical lattices

i

)( if

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)(

TTR

Single-particle conductivity: only states in the energy strip ~T near Fermi energy contribute

R(T)

Superconductivity: states in the energy strip ~ near the Fermi-energy contribute

)0()(~)( TRTRR c

R

)( cTR )( cTR

Interaction sets in a new scale which stays

constant as T->0

Weak and strong disorder

VW /

L

disorder

Critical states Localized states

Extended states

Anderson transition

F~

Multifractality of critical and off-critical states

)1(2 1|)(| nd

r

ni nLr

W>Wc W<Ec

Matrix elements

Interaction comes to play via matrix elements

)()( 22 rrrdVM mnd

nm

Ideal metal and insulator

22)()( rrrVdM mn

dnm

Metal: 111

VV

VV Small amplitude 100% overlap

Insulator:

111

VV

d

ddd

Large

amplitude but rare overlap

Critical enhancement of correlations

ddEE /1 2|'|

Amplitude higher than in a metal but almost

full overlap

States rather remote \E-E’|<E0) in energy are strongly correlated

Simulations on 3D Anderson model

Ideal metal: l

0

0E

dd

EE

E/1

0

2

'

Multifractal metal: l

0

WW

W

c

cF

mnmn

mnmnmn

d

EEEE

EEEErrrdV

EEC

,

22

,

)'()(

)'()()()(

)'(

cW

DE ~

13000

10

d Critical power law

persists

W=10

W=5

W=2

Wc=16.5

3/10 ~ cFW

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

);',()'(')( TrrKrdrr )'()'()()()();',( ** rrrrTUTrrK ijji

ijij

ji

jiij EE

TETET

)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?

(r) cannot be averaged

independently of K(r,r’) Fock space instead of the real space

FE FESuperconducting phase Normal phase

0, yxiS 0, yx

iS

)(2

jijiji

ijzi

iieff SSSSMUSH

Single-particle states, strong

disorder included

aaS )1(2

1

aaaaS z

Why the Fock-space mean field is better than the real-space one?

ijj j

jji M

E

TE

)2/tanh(

)()( 22 rrdrM jiij

)(2 yj

yi

xj

xi

jiij

zi

iieff SSSSMSH

Infinite or large coordination number for extended and weakly localized states

Weak fluctuations of Mij due to space integration

i j

MF critical temperature close to critical disorder

dd

EE

EEEM

EEME

TEEdEE

/1

02

|'|)'(

)'('

)2/'tanh()'(')(

78.10

)/1/(10 ~~ 2 EET dd

c

>> /1exp D

At a small parametrically

largeenhancement of Tc

M.V.Feigelman, L.B.Ioffe, V.E.K. and E.Yuzbshyan, Phys.Rev.Lett. v.98, 027001 (2007);

Thermodynamic phase fluctuations

drrrrrM lkjiijkl )()()()( Only possible if off-diagonal terms like

are taken into account

Ginzburg parameter

Gi~1

Expecting

1~/ cc TTCannot kill the parametrically

large enhancement

of Tc

The phase diagram

Mobility edge

Localized states

Extended states

Tc

(Disorder)

2.0BCS

BUT…

r

e2

Sweet life is only possible without Coulomb interaction

Virial expansion method of calculating the superconducting transition temperature

)()( cn

nc TT 1)(

)(lim 1

cn

cnn T

T

Replacing by: )()( 32 cc TT

Operational definition of Tc for numerical simulations

)(2 yj

yi

xj

xi

jiij

zi

iieff SSSSMUSH

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);

ddEcTcrit /1, 2/1

0

|'| EE

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

BUTFragile superconductivity:

Small fraction of superconducting phaseCritical current decreasing with disorder

Two-eigenfunction correlation in 3D Anderson model (insulator)

1ln)( dNC

Ideal insulator limit only in

one-dimensions

ddEE /1 2|'|

critical, multifractal

physics

Mott’s resonance

physics

Superconductor-Insulator transition: percolation without granulation

)(critTc

)(critTc

cT

cT

SC

INS

Only states in the strip ~Tc near the Fermi level take part in superconductivity

Coordination number K>>1

Coordination number K=0

First order transition?

ccc TUTMTK ~)()(

T

TTfTTT

TETfTKMU

dd )/()/(~

~)/()/()/(~

/00

0

2

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

ccc TUTMTK ~)()(

T

ddTTTTETKMU /000

2)/(~)/)(/(~

KMU

Corrections due to off-diagonal terms

ijklklkl

klij MTU )(

mkkllkmiikmkmill

lli MMUMTU

,

)(

Average value of the correction term

increases Tc

Average correction is small when

12

24 23

dd

dd

Melting of phase by disorder)()()( rrr jiij

ijij

In the diagonal approximation

)()2/tanh()( 2 rTEr iii

i

mkkllkmiikmkmill

lli MMUMTU

,

)(

The sign correlation <(r)(r’)> is perfect : solutions 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?

mkkllkmiikmkmil MMUQ

,

Stochastic term:

22ilil QQ

d2

>d/2

d2 >d/2 weak oscillations

d2 < d/2 strong oscillations but still too

small to support the glassy solution

22/212 ~~ 2 NQNQ ildd

ild2<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 Tc 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 d2 <d/2