1. Universality in ultra-cold fermionic atom gases

2. with S. Diehl, H.Gies, J.Pawlowski 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 Bose-Einstein condensate (BEC ) for low T Fermions with attractive interactions (molecules play no role ) : BCS – superfluidity at low T BCS – superfluidity at low T by condensation of Cooper pairs by condensation of Cooper pairs Crossover by Feshbach resonance Crossover by Feshbach resonance as a transition in terms of external magnetic field as a transition in terms of external magnetic field

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

5. Feshbach resonance H.Stoof

6. scattering length BCSBEC

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

8. chemical potential BEC BCS inverse scattering length

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

10. concentration c = a k F, a(B) : scattering length needs computation of density n=k F 3 /(3π 2 ) BCSBEC dilute dense T = 0 non- interacting Fermi gas non- interacting Bose gas

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

12. Quantum Monte Carlo different methods

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

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

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

18. problems

19. (1) bridge from microphysics to macrophysics

20. (2) different effective degrees of freedom microphysics : single atoms (+ molecules on BEC – side ) (+ 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 microphysics known coupling can be tuned coupling can be tuned for tests of theoretical methods these are important advantages as compared to solid state physics ! 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 Non-relativistic fermion systems with precision similar to particle physics ! similar to particle physics ! ( QCD with quarks ) ( QCD with quarks )

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

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

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

27. parameters detuning ν(B) detuning ν(B) Yukawa or Feshbach coupling h φ Yukawa or Feshbach coupling h φ

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

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

30. parameters Yukawa or Feshbach coupling h φ Yukawa or Feshbach coupling h φ scattering length a scattering length a broad resonance : h φ drops out broad resonance : h φ drops out

31. concentration c

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

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

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

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

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

38. rescaled action M drops out M drops out all quantities in units of k F, ε F if all quantities in units of k F, ε F 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 integrate out all quantum and thermal fluctuations quantum effective action quantum effective action generates full propagators and vertices generates full propagators and vertices richer structure than classical action richer structure than classical action

41. effective potential minimum determines order parameter condensate fraction Ω c = 2 ρ 0 /n

42. renormalized fields and couplings

43. results from functional renormalization group

44. T = 0 condensate fraction BCS BEC

45. T = 0 gap parameter BCSBEC Δ

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

47. T = 0 Yukawa coupling

48. temperature dependence of condensate

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

50. crossover phase diagram

51. shift of BEC critical temperature

52. correlation length ξ kFξ kFξ kFξ kF (T-T c )/T c three values of c

53. universality

54. universality for broad resonances for large Yukawa couplings h φ : only one relevant parameter c only one relevant parameter c all other couplings are strongly attracted to partial fixed points all other couplings are strongly attracted to partial fixed points macroscopic quantities can be predicted macroscopic quantities can be predicted in terms of c and T/ε F in terms of c and T/ε F ( in suitable range for c -1 ; density sets scale ) ( in suitable range for c -1 ; density sets scale )

55. universality for narrow resonances Yukawa coupling becomes additional parameter Yukawa coupling becomes additional parameter ( marginal coupling ) ( marginal coupling ) also background scattering important also background scattering important

56. bare molecule fraction (fraction of microscopic closed channel molecules ) not all quantities are universal not all quantities are universal bare molecule fraction involves wave function renormalization that depends on value of Yukawa coupling bare molecule fraction involves wave function renormalization that depends on value of Yukawa coupling B[G] Experimental points by Partridge et al. 6 Li

57. method

58. effective action includes all quantum and thermal fluctuations includes all quantum and thermal fluctuations formulated here in terms of renormalized fields formulated here in terms of renormalized fields involves renormalized couplings involves renormalized couplings

59. effective potential value of φ at potential minimum : value of φ at potential minimum : order parameter, determines condensate fraction order parameter, determines condensate fraction second derivative of U with respect to φ yields correlation length second derivative of U with respect to φ yields correlation length derivative with respect to σ yields density derivative with respect to σ yields density

60. functional renormalization group make effective action depend on scale k : make effective action depend on scale k : include only fluctuations with momenta larger than k include only fluctuations with momenta larger than k ( or with distance from Fermi-surface larger than k ) ( or with distance from Fermi-surface larger than k ) k large : no fluctuations, classical action k large : no fluctuations, classical action k → 0 : quantum effective action k → 0 : quantum effective action effective average action ( same for effective potential ) effective average action ( same for effective potential ) running couplings running couplings

61. microscope with variable resolution

62. running couplings : crucial for universality for large Yukawa couplings h φ : only one relevant parameter c only one relevant parameter c all other couplings are strongly attracted to partial fixed points all other couplings are strongly attracted to partial fixed points macroscopic quantities can be predicted macroscopic quantities can be predicted in terms of c and T/ε F in terms of c and T/ε F ( in suitable range for c -1 ) ( in suitable range for c -1 )

63. running potential micro macro here for scalar theory

64. physics at different length scales microscopic theories : where the laws are formulated microscopic theories : where the laws are formulated effective theories : where observations are made effective theories : where observations are made effective theory may involve different degrees of freedom as compared to microscopic theory effective theory may involve different degrees of freedom as compared to microscopic theory example: microscopic theory only for fermionic atoms, macroscopic theory involves bosonic collective degrees of freedom ( φ ) example: microscopic theory only for fermionic atoms, macroscopic theory involves bosonic collective degrees of freedom ( φ )

65. Functional Renormalization Group describes flow of effective action from small to large length scales perturbative renormalization : case where only couplings change, and couplings are small

66. conclusions the challenge of precision : substantial theoretical progress needed substantial theoretical progress needed “phenomenology” has to identify quantities that are accessible to precision both for experiment and theory “phenomenology” has to identify quantities that are accessible to precision both for experiment and theory dedicated experimental effort needed dedicated experimental effort needed

67. challenges for experiment study the simplest system study the simplest system identify quantities that can be measured with precision of a few percent and have clear theoretical interpretation identify quantities that can be measured with precision of a few percent and have clear theoretical interpretation precise thermometer that does not destroy probe precise thermometer that does not destroy probe same for density same for density

68. / Wegner, Houghton functional renormalization group

69. effective average action here only for bosons, addition of fermions straightforward

70. Flow equation for average potential + contribution from fermion fluctuations

71. Simple one loop structure – nevertheless (almost) exact

72. Infrared cutoff

73. Partial differential equation for function U(k,φ) depending on two variables Z k Z k = c k -η

74. Regularisation For suitable R k : Momentum integral is ultraviolet and infrared finite Momentum integral is ultraviolet and infrared finite Numerical integration possible Numerical integration possible Flow equation defines a regularization scheme ( ERGE –regularization ) Flow equation defines a regularization scheme ( ERGE –regularization )

75. Integration by momentum shells Momentum integral is dominated by q 2 ~ k 2. Flow only sensitive to physics at scale k

76. Wave function renormalization and anomalous dimension for Z k (φ,q 2 ) : flow equation is exact !

77. Flow of effective potential Ising model CO 2 T * =304.15 K p * =73.8.bar ρ * = 0.442 g cm-2 Experiment : S.Seide … Critical exponents

78. Critical exponents, d=3 ERGE world

79. Solution of partial differential equation : yields highly nontrivial non-perturbative results despite the one loop structure ! Example: Kosterlitz-Thouless phase transition

80. Exact renormalization group equation

81. end

82. Effective average action and exact renormalization group equation

83. Generating functional Generating functional

84. Loop expansion : perturbation theory with infrared cutoff in propagator Effective average action

85. Quantum effective action

86. Truncations Truncations Functional differential equation – cannot be solved exactly cannot be solved exactly Approximative solution by truncation of most general form of effective action most general form of effective action

87. Exact flow equation for effective potential Evaluate exact flow equation for homogeneous field φ. Evaluate exact flow equation for homogeneous field φ. R.h.s. involves exact propagator in homogeneous background field φ. R.h.s. involves exact propagator in homogeneous background field φ.

88. two body limit ( vacuum )