Multifractal superconductivity Vladimir Kravtsov ICTP Trieste Collaboration Michael

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 3 d 3 d lattice tight-binding model with diagonal disorder Local tunable attraction NO Coulomb interaction Relevant for cold atoms in disordered optical lattices

Cold atoms trapped in an optical lattice Fermionic atoms trapped in an optical lattice speckles Disorder is produced by: Other trapped atoms (impurities)

New possibilities for superconductivity in systems of cold atoms o o 1 d localization is experimentally observed [J. Billy et al. , Nature 453, 891, (2008); Roati Strong controllable disorder, et al. , Nature 453, 895 (2008)] realization of the is. Anderson 2 d and 3 d localization on the way model Tunable short-range interaction between atoms: superconductive properties vs. dimensionless attraction constant No long-range Coulomb interaction

Q: What does the strong disorder do to superconductivity? A 1: disorder gradually kills superconductivity A 2: eventually disorder kills superconductivity but before killing it enhances it

Why superconductivity is possible when singleparticle states are localized Single-particle conductivity: only states in the energy strip ~T near Fermi energy contribute Superconductivity: states in the energy strip ~D near the Fermi-energy contribute Interaction sets in a new scale D which stays constant as T->0 R(T) x R(D)

Weak and strong disorder Anderson transition disorder L Extended states Critical states Localized states

Multifractality of critical and off-critical states W>Wc W<Ec

Matrix elements Interaction comes to play via matrix elements

Ideal metal and insulator Metal: Insulator: Small amplitude 100% overlap Large amplitude but rare overlap

Critical enhancement of correlations Amplitude higher than in a metal but almost full overlap States rather remote (d<<E-E’|<E 0) in energy are strongly correlated

Simulations on 3 D Anderson model Wc=16. 5 W=10 Multifractal metal: x> l 0 Critical power law persists W=5 Ideal metal: x< l 0 W=2

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 1/V Wavefunctions drop out of the equation Anderson theorem: Tc does not depend on properties of wavefunctions

What to do at strong disorder? D(r) cannot be averaged independently of K(r, r’) Fock space instead of the real space Superconducting phase Normal phase Single-particle states, strong disorder included

Why the Fock-space mean field is better than the real-space one? i j Infinite or large coordination number for extended and weakly localized states Weak fluctuations of Mij due to space integration

MF critical temperature close to At a small l critical disorder parametrically large enhancement of Tc >> M. V. Feigelman, L. B. Ioffe, V. E. K. and E. Yuzbshyan, Phys. Rev. Lett. v. 98, 027001 (2007);

Thermodynamic phase fluctuations Only possible if off-diagonal terms like are taken into account Ginzburg parameter Gi~1 Expecting Cannot kill the parametrically large enhancement of Tc

The phase diagram Tc Mobility edge Extended states Localized states (Disorder)

BUT…

Sweet life is only possible without Coulomb interaction

Virial expansion method of calculating the superconducting transition temperature Replacing by: Operational definition of Tc for numerical simulations

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); MF result Virial expansion

2 d Analogue: the Mayekawa. Fukuyama-Finkelstein effect The diffuson diagrams The cooperon diagrams

Superconducting transition temperature Virial expansion on the 3 d Anderson model metal insulator Anderson localization transition (disorder)

Conclusion: Enhancement of Tc by disorder metal insulator 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 3 D Anderson model (insulator) Mott’s resonance physics Ideal insulator limit only in onedimensions critical, multifractal physics

Superconductor-Insulator transition: percolation without granulation Coordination number K>>1 Coordination number K=0 SC INS Only states in the strip ~Tc near the Fermi level take part in superconductivity

First order transition? T Tc T f(x)->0 at x<<1

Conclusion o o o 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.

T M IN BCS limit SC SC or IN SC disorder Anderson localization transition KMU T T

Corrections due to off-diagonal terms Average value of the correction term increases Tc Average correction is small when

Melting of phase by disorder In the diagonal approximation The sign correlation <D(r)D(r’)> is perfect : solutions Di>0 do not lead to a global phase destruction Beyond the diagonal approximation: stochastic term destroys phase correlation

How large is the stochastic term? Stochastic term: d 2<d/2 d 2 >>d/2 weak oscillations d d 2 < d/2 strong oscillations / but still too small to support 2 the glassy solution

More research is needed

Conclusions o o o 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 d 2 <d/2