Universality in ultra-cold fermionic atom gases

1. Universality in ultra-cold fermionic atom gases

2. Universality in ultra-cold fermionic atom gases with S. Diehl , H.Gies , J.Pawlowski

3. BEC – BCS crossover Bound molecules of two atoms on microscopic scale: Bose-Einstein condensate (BEC ) for low T Fermions with attractive interactions (molecules play no role ) : BCS – superfluidity at low T by condensation of Cooper pairs Crossoverby Feshbach resonance as a transition in terms of external magnetic field

4. microphysics • determined by interactions between two atoms • length scale : atomic scale

5. Feshbach resonance H.Stoof

6. scattering length BEC BCS

7. many body physics • dilute gas of ultra-cold atoms • length scale : distance between atoms

8. chemical potential BCS BEC inverse scattering length

9. BEC – BCS crossover • qualitative and partially quantitative theoretical understanding • mean field theory (MFT ) and first attempts beyond concentration : c = a kF reduced chemical potential : σ˜= μ/εF Fermi momemtum : kF Fermi energy : εF binding energy : T = 0 BCS BEC

10. concentration • c = a kF , a(B) : scattering length • needs computation of density n=kF3/(3π2) dilute dense dilute non- interacting Fermi gas non- interacting Bose gas T = 0 BCS BEC

11. universality same curve for Li and K atoms ? dilute dense dilute T = 0 BCS BEC

12. different methods Quantum Monte Carlo

13. who cares about details ? a theorists game …? MFT RG

14. a theorists dream : reliable method for strongly interacting fermions “ solving fermionic quantum field theory “

15. experimental precision tests are crucial !

16. precision many body theory- quantum field theory - so far : • particle physics : perturbative calculations magnetic moment of electron : g/2 = 1.001 159 652 180 85 ( 76 ) ( Gabrielse et al. ) • statistical physics : universal critical exponents for second order phase transitions : ν = 0.6308 (10) renormalization group • lattice simulations for bosonic systems in particle and statistical physics ( e.g. QCD )

17. QFT with fermions needed: universal theoretical tools for complex fermionic systems wide applications : electrons in solids , nuclear matter in neutron stars , ….

18. problems

19. (1) bridge from microphysics tomacrophysics

20. (2) different effective degrees of freedom microphysics : single atoms (+ molecules on BEC – side ) macrophysics : bosonic collective degrees of freedom compare QCD : from quarks and gluons to mesons and hadrons

21. (3) no small coupling

22. ultra-cold atoms : • microphysics known • coupling can be tuned • for tests of theoretical methods these are important advantages as compared to solid state physics !

23. challenge for ultra-cold atoms : Non-relativistic fermion systems with precision similar to particle physics ! ( QCD with quarks )

24. functional renormalization group • conceived to cope with the above problems • should be tested by ultra-cold atoms

25. QFT for non-relativistic fermions • functional integral, action perturbation theory: Feynman rules τ : euclidean time on torus with circumference 1/T σ : effective chemical potential

26. variables • ψ : Grassmann variables • φ : bosonic field with atom number two What is φ ? microscopic molecule, macroscopic Cooper pair ? All !

27. parameters • detuning ν(B) • Yukawa or Feshbach coupling hφ

28. fermionic action equivalent fermionic action , in general not local

29. scattering length a • broad resonance : pointlike limit • large Feshbach coupling a= M λ/4π

30. parameters • Yukawa or Feshbach coupling hφ • scattering length a • broad resonance : hφdrops out

31. concentration c

33. universality • Are these parameters enough for a quantitatively precise description ? • Have Li and K the same crossover when described with these parameters ? • Long distance physics looses memory of detailed microscopic properties of atoms and molecules ! universality for c-1 = 0 : Ho,…( valid for broad resonance) here: whole crossover range

34. analogy with particle physics microscopic theory not known - nevertheless “macroscopic theory” characterized by a finite number of “renormalizable couplings” me , α ; g w , g s , M w , … here : c , hφ( only c for broad resonance )

35. analogy with universal critical exponents only one relevant parameter : T - Tc

36. universality • issue is not that particular Hamiltonian with two couplings ν , hφgives good approximation to microphysics • large class of different microphysical Hamiltonians lead to a macroscopic behavior described only by ν , hφ • difference in length scales matters !

37. units and dimensions • c = 1 ; ħ =1 ; kB = 1 • momentum ~ length-1 ~ mass ~ eV • energies : 2ME ~ (momentum)2 ( M : atom mass ) • typical momentum unit : Fermi momentum • typical energy and temperature unit : Fermi energy • time ~ (momentum) -2 • canonical dimensions different from relativistic QFT !

38. rescaled action • M drops out • all quantities in units of kF , εF if

39. what is to be computed ? Inclusion of fluctuation effects via functional integral leads to effective action. This contains all relevant information for arbitrary T and n !

40. effective action • integrate out all quantum and thermal fluctuations • quantum effective action • generates full propagators and vertices • richer structure than classical action

41. effective potential minimum determines order parameter condensate fraction Ωc = 2 ρ0/n

42. renormalized fields and couplings

43. resultsfromfunctional renormalization group

44. condensate fraction T = 0 BCS BEC

45. gap parameter Δ T = 0 BCS BEC

46. limits BCS for gap Bosons with scattering length 0.9 a

47. Yukawa coupling T = 0

48. temperature dependence of condensate

49. condensate fraction :second order phase transition c -1 =1 free BEC c -1 =0 universal critical behavior T/Tc

50. crossover phase diagram