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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. - PowerPoint PPT Presentation
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