Multifractal superconductivity

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

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

4. Strong controllable disorder, realization of the Anderson model 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

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

7. Weak and strong disorder Anderson transition disorder L Extendedstates Critical states Localizedstates

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

9. Matrix elements Interaction comes to play via matrix elements

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

11. Critical enhancement of correlations Amplitude higher than in a metal but almost full overlap States rather remote (d<<\E-E’|<E0) in energy are strongly correlated

12. Simulations on 3D Anderson model Wc=16.5 Multifractal metal: x> l 0 W=10 Critical power law persists W=5 W=2 Ideal metal:x< l 0

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

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

16. i j 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 Mij due to space integration

17. >> MF critical temperature close to critical disorder At a small l parametrically largeenhancement of Tc 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 Cannot kill the parametrically large enhancement of Tc Expecting Ginzburg parameter Gi~1

19. The phase diagram Mobility edge Tc Extended states Localized states (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); MF result Virial expansion

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 metal insulator Anderson localization transition (disorder)

26. 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

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

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

29. First order transition? Tc T T 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. T T T M IN BCS limit SC SC or IN SC disorder Anderson localization transition KMU

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

33. 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

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

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 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