1. Multifractal superconductivity Vladimir Kravtsov, ICTP (Trieste) Collaboration: Michael Feigelman (Landau Institute) Emilio Cuevas (University of Murcia) Lev Ioffe (Rutgers) Seattle, August 27, 2009

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

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

4. 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

5. 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

6. Why superconductivity is possible when single- particle states are localized 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 R Interaction sets in a new scale which stays constant as T->0

7. Weak and strong disorder L disorder Critical statesLocalized states Extended states Anderson transition

8. Multifractality of critical and off-critical states W>W c W<E c

9. Matrix elements Interaction comes to play via matrix elements

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

11. Critical enhancement of correlations Amplitude higher than in a metal but almost full overlap States rather remote  \ E-E’|<E 0 ) in energy are strongly correlated

12. Simulations on 3D Anderson model Ideal metal:  l 0 Multifractal metal:  l 0 Critical power law persists W=10 W=5 W=2 W c =16.5

13. 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?

14. Mean-field approximation and the Anderson theorem Wavefunctions drop out of the equation Anderson theorem: Tc does not depend on properties of wavefunctions 1/V

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

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

17. MF critical temperature close to critical disorder >> 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);

18. 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

19. The phase diagram Mobility edge Localized states Extended states TcTc (Disorder)

20. BUT…

21. Sweet life is only possible without Coulomb interaction

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

23. 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); Virial expansion MF result

24. 2d Analogue: the Mayekawa- Fukuyama-Finkelstein effect The diffuson diagrams The cooperon diagrams

25. Superconducting transition temperature Virial expansion on the 3d Anderson model metalinsulator (disorder) Anderson localization transition

26. metalinsulator 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

27. Two-eigenfunction correlation in 3D Anderson model (insulator) Ideal insulator limit only in one-dimensions  critical, multifractal physics Mott’s resonance physics

28. Superconductor-Insulator transition: percolation without granulation 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

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

30. 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.

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

32. Corrections due to off-diagonal terms Average value of the correction term increases T c Average correction is small when

33. Melting of phase by disorder In the diagonal approximation The sign correlation is perfect : solutions  i>0 do not lead to a global phase destruction stochastic term destroys phase correlation Beyond the diagonal approximation:

34. How large is the stochastic term? Stochastic term: d2>d/2d2>d/2 d 2 >d/2 weak oscillations d 2 < d/2 strong oscillations but still too small to support the glassy solution d 2 <d/2

35. More research is needed

36. 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