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

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

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

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

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

6. Relevant Scales  Mean level spacing Δ 0 Superconducting gap  F Fermi Energy L typical length l coherence length ξ Superconducting coherence length Conditions BCS  / Δ 0 << 1 Semiclassical 1/k F L << 1 Quantum coherence l >> L ξ >> L For Al the optimal region is L ~ 10nm

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

8. A little history 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 T c Olofsson (2008): Estimation of fluctuations in BCS, no correlations Superconductivity in particular geometries

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

10.  / Δ 0 >> 1 We are in business! systematic No systematic BCS treatment of the dependence of size and shape

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

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 f(L,  -  ’,  F ) is a simple function For l>>L ergodic theorems assures universality

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

15. Now it is easy

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

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

18. 3d integrable Numerical & analyticalCube & rectangle

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

20. Mesoscopic corrections versus corrections to mean field Finite size corrections to BCS Matveev-LarkinPair 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 arXiv:0904.0354v1 Sorry but in Pb only small fluctuations Are you 300% sure?

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

23. 0 nm 7 nm

24. dI/dV

25. + Theory

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

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

28. Dynes fitting Problem:  > 

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

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

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

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

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

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

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

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 1. Microscopic Hamiltonian is not important 2. Large N approximation OK Why?

41. 1. d=2 and AdS 4 geometry 2. For c 3 = c 4 = 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 Holographic approach to phase transitions Phys. Rev. D 81, 041901 (2010)

42. For c 4 > 1 or c 3 > 0 the transition becomes first order A jump in the condensate at the critical temperature is clearly observed for c 4 > 1 The discontinuity for c 4 > 1 is a signature of a first order phase transition. Results I

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

44. The spectroscopic gap becomes larger and the coherence peak narrower as c 4 increases. Results III

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. Unitarity regime and Efimov states 3 identical bosons with a large scattering length a 1/a1/a Energy trimer 3 particles Ratio = 514 Efimov trimers Naidon, Tokyo Bound states exist even for repulsive interactions! Predicted by V. Efimov in 1970 Form an infinite series (scale invariance) Bond is purely quantum- mechanical

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

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

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

51. What next? Quantum Fermi gases From few-body to many-body Discovery of new forms of quantum matter Relation to high Tc superconductivity

52. 1. A condensate that is non zero at low T and that vanishes at a certain T = T c 2. It is possible to study different phase transitions 3. A string theory embedding is known Holographic approach to phase transitions Phys. Rev. D 81, 041901 (2010) A U(1) field , p scalars F Maxwell tensor

53. E. Yuzbashyan, Rutgers B. Altshuler Columbia JD Urbina Regensburg S. Bose Stuttgart M. Tezuka Kyoto S. Franco, Santa Barbara K. Kern, Stuttgart J. Wang Singapore D. Rodriguez Queen Mary K. Richter Regensburg Let’s do it!! P. Naidon Tokyo