A theory of finite size effects in BCS superconductors: The making of a paper Antonio M. García-García Princeton and ICTP Phys. Rev. Lett. 100, (2008), AGG, Urbina, Yuzbashyan, Richter, Altshuler. YuzbashyanAltshuler Urbina Richter

L 1. How do the properties of a clean BCS superconductor depend on its size and shape? 2. To what extent are these results applicable to realistic grains? Main goals

Talk to Emil Quantum chaos, trace formula…what? Richardson equations, Anderson representation …what? Princeton 2005: A false start Superconductivity?, Umm, semiclassical, fine Superconductivity, spin, semiclassical

Spring 2006: A glimmer of hope Semiclassical: To express quantum observables in terms of classical quantities. O nly 1/k F L <<1, Berry, Gutzwiller, Balian, Bloch Gutzwiller trace formula Can I combine this? Is it already done?

Non oscillatory terms Oscillatory terms in terms of classical quantities only Semiclassical (1/k F L >> 1) expression of the spectral density,Gutzwiller, Berry

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

A little history 1959, Anderson: superconductor if  / Δ 0 > 1? 1962, 1963, Parmenter, Blatt Thompson. BCS in a cubic grain 1972, Muhlschlegel, thermodynamic properties 1995, Tinkham experiments with Al grains ~ 5nm 2003, Heiselberg, pairing in harmonic potentials 2006, Shanenko, Croitoru, BCS in a wire 2006 Devreese, Richardson equation in a box 2006, Kresin, Boyaci, Ovchinnikov, Spherical grain, high T c 2008, Olofsson, fluctuations in Chaotic grains, no matrix elements!

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

Fall 06: Hitting a bump 3d cubic Al grain 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

With help we could achieve it Winter 2006: From desperation to hope ?

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

Semiclassical (1/k F L >> 1) expression of the matrix elements valid for l >> L!! Technically is much more difficult because it involves the evaluation of all closed orbits not only periodic ω =  -  ’ A few months later This result is relevant in virtually any mean field approach

Non oscillatory terms Oscillatory terms in terms of classical quantities only Semiclassical (1/k F L >> 1) expression of the spectral density,Gutzwiller, Berry

Expansion in powers of  /  0 and 1/k F L 2d chaotic and rectangular 3d chaotic and rectangular Summer 2007

3d chaotic The sum over g(0) is cut-off by the coherence length ξ Universal function Importance of boundary conditions

3d chaotic AL grain k F = 17.5 nm -1  = 7279/N mv  0 = 0.24mv From top to bottom: 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 In this range of parameters the leading correction to the gap comes from of the matrix elements not the spectral density

2d chaotic Importance of Matrix elements!! Universal function Importance of boundary conditions

2d chaotic AL grain k F = 17.5 nm -1  = 7279/N mv  0 = 0.24mv From top to bottom: L = 6nm, Dirichlet,  /Δ 0 =0.77 L= 6nm, Neumann,  /Δ 0, =0.77 L = 8nm, Dirichlet,  /Δ 0 =0.32 L = 10nm, Dirichlet,  /Δ 0,= 0.08 In this range of parameters the leading correction to the gap comes from of the matrix elements not the spectral density

3d integrable V = n/181 nm -3 Numerical & analyticalCube & parallelepiped No role of matrix elements Similar results were known in the literature from the 60’s Fall 2007, sent to arXiv!

Spatial Dependence of the gap The prefactor suppresses exponentially the contribution of eigenstates with energy > Δ 0 The average is only over a few eigenstates around the Fermi surface Maybe some structure is preserved

N = 2998

Scars N =4598 Anomalous enhancement of the quantum probability around certain unstable periodic orbits (Kaufman, Heller) N =5490 Experimental detection possible (Yazdani) No theory so trial and error

Is this real? Real (small) Grains Coulomb interactions Phonons Deviations from mean field Decoherence Geometrical deviations No Yes

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

Decoherence and geometrical deformations Decoherence effects and small geometrical deformations in otherwise highly symmetric grains weaken mesoscopic effects How much? To what extent are our previous results robust? Both effects can be accounted analytically by using an effective cutoff in the semiclassical expressions

D(L p /l) The form of the cutoff depends on the mechanism at work Finite temperature,Leboeuf Random bumps, Schmit,Pavloff Multipolar corrections, Brack,Creagh

Fluctuations are robust provided that L >> l Non oscillating deviations present even for L ~ l

The Future?

1. Disorder and finite size effects in superconductivity 2. AdS-CFT techniques in condensed matter physics Control of superconductivity (T c ) What? Why? Superconductivity 1. New high T c superconducting materials 2. Control of interactions and disorder in cold atoms 3. New analytical tools Why now? 4.Better exp control in condensed matter

arXiv: v1

Test of localization by Cold atoms Finite size/disorder effects in effects insuperconductivity GOALS Comparison with superconducting grains exp. Numerical and theoretical analysis of experimental speckle potentials Comparison with experiments (cold atoms)‏ Mean field region Semiclassical + known many body techniques Comparison with exp. blackbody Semiclassical techniques plus Stat. Mech. results IDEA THEORY REALITY CHECK Exp. verification of localization Bad Good Mesoscopicstatistical mechanics mechanics Great! Superconducting circuits with higher critical temperature Qualitiy control manufacturedcavities Test of quantum mechanics E. Yuzbashian, J. Urbina, B. Altshuler. D. Rodriguez Wang Jiao S. Sinha, E. Cuevas 053 Time (years)‏ EasyMediumDifficultMilestone Strong Coupling AdS -CFT techniques Great! Comparison cold atoms experiments Test Ergodic Hyphothesis Numerics + beyond semiclassical tech. Novel states quantum matter Novel states quantum matter Great! Comparison BEC-BCS physics Theory of strongly interacting fermions