An excursion into modern superconductivity: from nanoscience to cold atoms and holography

1. An excursion into modern superconductivity: from nanoscience to cold atoms and holography Antonio M. García-García Sangita Bose, Tata, Max Planck Stuttgart Diego Rodriguez Queen Mary Masaki Tezuka Kyoto Yuzbashyan Rutgers Altshuler Columbia Jiao Wang NUS Richter Regensburg Urbina Regensburg Kern Stuttgart Sebastian Franco Santa Barbara

2. Superconductivity in nanograins New forms of superconductivity Practical Theoretical Superconductivity Increasing the superconductor Tc Technical New tools String Theory

3. Enhancement and control of superconductivity in nanograins Phys. Rev. Lett. 100, 187001 (2008) Sangita Bose, Tata, Max Planck Stuttgart Yuzbashyan Rutgers Altshuler Columbia arXiv:0911.1559 Nature Materials Kern Ugeda, Brihuega Richter Regensburg Urbina Regensburg

4. Main goals 1. Analytical description of a clean, finite-size BCS superconductor? 2. Are these results applicable to realistic grains? 3. Is it possible to increase the critical temperature? L

5. The problem Semiclassical 1/kF L <<1Berry, Gutzwiller, Balian BCS gap equation V bulk Δ~ De-1/ V finite Δ=? ? Can I combine this? Is it already done?

6. Relevant Scales Δ0 Superconducting gap L typical length  Mean level spacing l coherence length ξSuperconducting coherence length F Fermi Energy Conditions BCS / Δ0 << 1 Semiclassical1/kFL << 1 Quantum coherence l >> L ξ >> L For Al the optimal region is L ~ 10nm

7. Maybe it is possible Go ahead! This has not been done before It is possible but, is it relevant? Corrections to BCS smaller or larger? If so, in what range of parameters? Let’s think about this

8. A little history Superconductivity in particular geometries Parmenter, Blatt, Thompson (60’s) : BCS in a rectangular grain Heiselberg (2002): BCS in harmonic potentials, cold atom appl. Shanenko, Croitoru (2006): BCS in a wire Devreese (2006): Richardson equations in a box Kresin, Boyaci, Ovchinnikov (2007) Spherical grain, high Tc Olofsson (2008): Estimation of fluctuations in BCS, no correlations

9. Nature of superconductivity (?) in ultrasmall systems Breaking of superconductivity for / Δ0 > 1?Anderson (1959) Estimation. No rigorous! Thermodynamic properties Muhlschlegel, Scalapino (1972) Description beyond BCS Experiments Tinkhamet al. (1995) . Guo et al., Science 306, 1915, Superconductivity Modulated by quantum Size Effects. Even for / Δ0 ~ 1 there is “supercondutivity 1.Richardson’s equations: Good but Coulomb, phonon spectrum? 2.BCS fine until / Δ0 ~ 2 T = 0 and / Δ0 > 1 (1995-) Richardson, von Delft, Braun, Larkin, Sierra, Dukelsky, Yuzbashyan

10. / Δ0 >> 1 No systematic BCS treatment of the dependence of size and shape We are in business!

11. Hitting a bump λ/V ? In,n should admit a semiclassical expansion but how to proceed? For the cube yes but for a chaotic grain I am not sure I ~1/V? Fine, but the matrix elements?

12. Yes, with help, we can From desperation to hope ?

13. Regensburg, we have got a problem!!! Do not worry. It is not an easy job but you are in good hands Nice closed results that do not depend on the chaotic cavity For l>>L ergodic theorems assures universality f(L,- ’, F) is a simple function

14. A few months later Semiclassical (1/kFL >> 1) expression of the matrix elements valid for l >> L!! ω = -’ Relevant in any mean field approach with chaotic one body dynamics

15. Now it is easy

16. 3d chaotic Sum is cut-off ξ Boundary conditions Universal function Enhancement of SC!

17. 3d chaotic Al grain kF = 17.5 nm-1  = 7279/N mV 0 = 0.24mV L = 6nm, Dirichlet, /Δ0=0.67 L= 6nm, Neumann, /Δ0,=0.67 L = 8nm, Dirichlet, /Δ0=0.32 L = 10nm, Dirichlet, /Δ0,= 0.08 For L< 9nm leading correction comes from I(,’)

18. 3d integrable Numerical & analytical Cube & rectangle

19. From theory to experiments L ~ 10 nm Sn, Al… Is it taken into account? Real (small) Grains No, but screening should be effective Coulomb interactions No, but no strong effect expected Surface Phonons Deviations from mean field Yes Decoherence Yes Fluctuations No

20. Mesoscopic corrections versus corrections to mean field Finite size corrections to BCS Matveev-Larkin Pair breaking Janko,1994 The leading mesoscopic corrections contained in (0) are larger The correction to (0) proportional to  has different sign

21. Experimentalists are coming Sorry but in Pb only small fluctuations Are you 300% sure? arXiv:0904.0354v1

22. However in Sn is very different !!!!!!!!!!!!!!!!!!!!!!!!!!!!! Pb and Sn are very different because their coherence lengths are very different.

23. 7 nm 0 nm

24. dI/dV

25. Theory +

26. Direct observation of thermal fluctuations and the gradual breaking of superconductivity in single, isolated Pbnanoparticles ? Pb

27. Theoretical description of dI/dV ? dI/dV Dynes formula Solution Thermal fluctuations + BCS Finite size effects + Deviations from mean field

28. Dynes fitting Problem:  > 

29. How? Finite T Thermal fluctuations Static Path approach BCS finite size effects Part I Deviations from BCS Richardson formalism No quantum fluctuations!

31. T=0 BCS finite size effects Part I Deviations from BCS Richardson formalism Altshuler, Yuzbashyan, 2004 No quantum fluctuations! Not important h ~ 6nm

32. Cold atom physics and novel forms of superconductivity Temperatures can be lowered up to the nano Kelvin scale Ideal laboratory to test quantum phenomena Cold atoms settings Interactions can be controlled by Feshbachresonances Until 2005

33. 2005 - now 1. Disorder & magnetic fields Test of Anderson localization, Hall Effect 2. Non-equilibrium effects Test ergodicity hypothesis 3. Efimov physics Bound states of three quantum particles do exist even if interactions are repulsive

34. What is the effect of disorder in 1d Fermi gases? Stability of the superfluid state in a disordered 1D ultracoldfermionic gas Masaki Tezuka (U. Tokyo), Antonio M. Garcia-Garcia arXiv:0912.2263 DMRG analysis of Why?

35. Speckel potential Our model!! speckle incommensurate lattice pure random with correlations quasiperiodic localization for any D localization transition at finite D = 2 Only two types of disorder can be implemented experimentally Modugno

36. Results I Attractive interactions enhance localization U = 1 c = 1<2

37. Results II Weak disorder enhances superfluidity

38. Results III A pseudo gap phase exists. Metallic fluctuations break long range order Results IV Spectroscopic observables are not related to long range order

39. String theory meets condensed matter Strongly coupled field theory A solution looking for a problem Weakly coupled gravity dual N=4 Super-Yang Mills CFT Anti de Sitter space AdS Applications in high Tc superconductivity Why? Powerful tool to deal with strong interactions Why now? New field. Potential for high impact JHEP 1004:092 (2010) What is next? Transition from qualitative to quantitative Phys. Rev. D 81, 041901 (2010) Collaboration with string theorists

40. Problems 1. Estimation of the validity of the AdS-CFT approach 2. Large N limit For what condensed matter systems these problems are minimized? Phase Transitions triggered by thermal fluctuations Why? 1. Microscopic Hamiltonian is not important 2. Large N approximation OK

41. Holographic approach to phase transitions Phys. Rev. D 81, 041901 (2010) 1. d=2 and AdS4 geometry 2. For c3 = c4 = 0 mean field results 3. Gauge field A is U(1) and  is a scalar 4. A realization in string theory and M theory is known for certain choices of ƒ 5. By tuning ƒ we can reproduce many types of phase transitions

42. Results I For c4 > 1 or c3 > 0 the transition becomes first order A jump in the condensate at the critical temperature is clearly observed for c4 > 1 The discontinuity for c4 > 1 is a signature of a first order phase transition.

43. Second order phase transitions with non mean field critical exponents different are also accessible Results II 1. For c3 < -1 2. For Condensate for c = -1 and c4 = ½. β= 1, 0.80, 0.65, 0.5 for  = 3, 3.25, 3.5, 4, respectively

44. Results III The spectroscopic gap becomes larger and the coherence peak narrower as c4 increases.

45. Future 1. Extend results to β <1/2 2. Adapt holographic techniques to spin discrete 3. Effect of phase fluctuations. Mermin-Wegner theorem? 4. Relevance in high temperature superconductors

46. THANKS!

47. trimer Unitarity regime and Efimov states 3 identical bosons with a large scattering length a Efimov trimers Energy Bound states exist even for repulsive interactions! 3 particles 1/a Predicted by V. Efimov in 1970 trimer Bond is purely quantum- mechanical Ratio = 514 trimer Form an infinite series (scale invariance) Naidon, Tokyo

48. What would I bring to SeoulNationalUniversity? Expertise in interesting problems in condensed matter theory Cross disciplinary profile and interests with the common thread of superconductivity Teaching and leadership experience from a top US university Collaborators

49. Decoherence and geometrical deformations Decoherence effects and small geometrical deformations weaken mesoscopic effects How much? Both effects can be accounted analytically by using an effective cutoff in the trace formula for the spectral density To what extent is our formalism applicable?

50. Our approach provides an effective description of decoherence Non oscillating deviations present even for L ~ l