1. Functional renormalization group for the effective average action

2. wide applications particle physics gauge theories, QCD gauge theories, QCD Reuter,…, Marchesini et al, Ellwanger et al, Litim, Pawlowski, Gies,Freire, Morris et al., many others Reuter,…, Marchesini et al, Ellwanger et al, Litim, Pawlowski, Gies,Freire, Morris et al., many others electroweak interactions, gauge hierarchy problem electroweak interactions, gauge hierarchy problem Jaeckel,Gies,… Jaeckel,Gies,… electroweak phase transition electroweak phase transition Reuter,Tetradis,…Bergerhoff, Reuter,Tetradis,…Bergerhoff,

3. wide applications gravity asymptotic safety asymptotic safety Reuter, Lauscher, Schwindt et al, Percacci et al, Litim, Fischer Reuter, Lauscher, Schwindt et al, Percacci et al, Litim, Fischer

4. wide applications condensed matter unified description for classical bosons unified description for classical bosons CW, Tetradis, Aoki, Morikawa, Souma, Sumi, Terao, Morris, Graeter, v.Gersdorff, Litim, Berges, Mouhanna, Delamotte, Canet, Bervilliers, CW, Tetradis, Aoki, Morikawa, Souma, Sumi, Terao, Morris, Graeter, v.Gersdorff, Litim, Berges, Mouhanna, Delamotte, Canet, Bervilliers, Hubbard model Baier,Bick,…, Metzner et al, Salmhofer et al, Honerkamp et al, Krahl, Hubbard model Baier,Bick,…, Metzner et al, Salmhofer et al, Honerkamp et al, Krahl, disordered systems Tissier, Tarjus,Delamotte, Canet disordered systems Tissier, Tarjus,Delamotte, Canet

5. wide applications condensed matter equation of state for CO 2 Seide,… equation of state for CO 2 Seide,… liquid He 4 Gollisch,… and He 3 Kindermann,… liquid He 4 Gollisch,… and He 3 Kindermann,… frustrated magnets Delamotte, Mouhanna, Tissier frustrated magnets Delamotte, Mouhanna, Tissier nucleation and first order phase transitions nucleation and first order phase transitions Tetradis, Strumia,…, Berges,… Tetradis, Strumia,…, Berges,…

6. wide applications condensed matter crossover phenomena Bornholdt, Tetradis,… crossover phenomena Bornholdt, Tetradis,… superconductivity ( scalar QED 3 ) superconductivity ( scalar QED 3 ) Bergerhoff, Lola, Litim, Freire,… Bergerhoff, Lola, Litim, Freire,… non equilibrium systems non equilibrium systems Delamotte, Tissier, Canet, Pietroni Delamotte, Tissier, Canet, Pietroni

7. wide applications nuclear physics effective NJL- type models effective NJL- type models Ellwanger, Jungnickel, Berges, Tetradis,…, Pirner, Schaefer, Wambach, Kunihiro, Schwenk, Ellwanger, Jungnickel, Berges, Tetradis,…, Pirner, Schaefer, Wambach, Kunihiro, Schwenk, di-neutron condensates di-neutron condensates Birse, Krippa, Birse, Krippa, equation of state for nuclear matter equation of state for nuclear matter Berges, Jungnickel …, Birse, Krippa Berges, Jungnickel …, Birse, Krippa

8. wide applications ultracold atoms Feshbach resonances Diehl, Gies, Pawlowski,…, Krippa, Feshbach resonances Diehl, Gies, Pawlowski,…, Krippa, BEC Blaizot, Wschebor, Dupuis, Sengupta BEC Blaizot, Wschebor, Dupuis, Sengupta

9. unified description of scalar models for all d and N

10. Flow equation for average potential

11. Scalar field theory Scalar field theory

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

13. Infrared cutoff

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

15. Scaling form of evolution equation Tetradis … On r.h.s. : neither the scale k nor the wave function renormalization Z appear explicitly. Scaling solution: no dependence on t; corresponds to second order phase transition.

16. unified approach choose N choose N choose d choose d choose initial form of potential choose initial form of potential run ! run !

17. 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

18. Critical exponents, d=3 ERGE world

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

20. Essential scaling : d=2,N=2 Flow equation contains correctly the non- perturbative information ! Flow equation contains correctly the non- perturbative information ! (essential scaling usually described by vortices) (essential scaling usually described by vortices) Von Gersdorff …

21. Kosterlitz-Thouless phase transition (d=2,N=2) Correct description of phase with Goldstone boson Goldstone boson ( infinite correlation length ) ( infinite correlation length ) for T<T c for T<T c

22. Running renormalized d-wave superconducting order parameter κ in doped Hubbard model κ - ln (k/Λ) TcTc T>T c T<T c C.Krahl,… macroscopic scale 1 cm

23. Renormalized order parameter κ and gap in electron propagator Δ in doped Hubbard model 100 Δ / t κ T/T c jump

24. Temperature dependent anomalous dimension η T/T c η

25. convergence and errors for precise results: systematic derivative expansion in second order in derivatives for precise results: systematic derivative expansion in second order in derivatives includes field dependent wave function renormalization Z(ρ) includes field dependent wave function renormalization Z(ρ) fourth order : similar results fourth order : similar results apparent fast convergence : no series resummation apparent fast convergence : no series resummation rough error estimate by different cutoffs and truncations rough error estimate by different cutoffs and truncations

26. including fermions : no particular problem ! no particular problem !

27. changing degrees of freedom

28. Antiferromagnetic order in the Hubbard model A functional renormalization group study T.Baier, E.Bick, …

29. Hubbard model Functional integral formulation U > 0 : repulsive local interaction next neighbor interaction External parameters T : temperature μ : chemical potential (doping )

30. Fermion bilinears Introduce sources for bilinears Functional variation with respect to sources J yields expectation values and correlation functions

31. Partial Bosonisation collective bosonic variables for fermion bilinears collective bosonic variables for fermion bilinears insert identity in functional integral insert identity in functional integral ( Hubbard-Stratonovich transformation ) ( Hubbard-Stratonovich transformation ) replace four fermion interaction by equivalent bosonic interaction ( e.g. mass and Yukawa terms) replace four fermion interaction by equivalent bosonic interaction ( e.g. mass and Yukawa terms) problem : decomposition of fermion interaction into bilinears not unique ( Grassmann variables) problem : decomposition of fermion interaction into bilinears not unique ( Grassmann variables)

32. Partially bosonised functional integral equivalent to fermionic functional integral if Bosonic integration is Gaussian or: solve bosonic field equation as functional of fermion fields and reinsert into action

33. fermion – boson action fermion kinetic term boson quadratic term (“classical propagator” ) Yukawa coupling

34. source term is now linear in the bosonic fields

35. Mean Field Theory (MFT) Evaluate Gaussian fermionic integral in background of bosonic field, e.g.

36. Effective potential in mean field theory

37. Mean field phase diagram μ μ TcTc TcTc for two different choices of couplings – same U !

38. Mean field ambiguity TcTc μ mean field phase diagram U m = U ρ = U/2 U m = U/3,U ρ = 0 Artefact of approximation … cured by inclusion of bosonic fluctuations J.Jaeckel,…

39. Rebosonization and the mean field ambiguity

40. Bosonic fluctuations fermion loops boson loops mean field theory

41. Rebosonization adapt bosonization to every scale k such that adapt bosonization to every scale k such that is translated to bosonic interaction is translated to bosonic interaction H.Gies, … k-dependent field redefinition absorbs four-fermion coupling

42. Modification of evolution of couplings … Choose α k such that no four fermion coupling is generated Evolution with k-dependent field variables Rebosonisation

43. …cures mean field ambiguity TcTc U ρ /t MFT Flow eq. HF/SD

44. Flow equation for the Hubbard model T.Baier, E.Bick, …,C.Krahl

45. Truncation Concentrate on antiferromagnetism Potential U depends only on α = a 2

46. scale evolution of effective potential for antiferromagnetic order parameter boson contribution fermion contribution effective masses depend on α ! gap for fermions ~α

47. running couplings

48. unrenormalized mass term Running mass term four-fermion interaction ~ m -2 diverges -ln(k/t)

49. dimensionless quantities renormalized antiferromagnetic order parameter κ

50. evolution of potential minimum -ln(k/t) U/t = 3, T/t = 0.15 κ 10 -2 λ

51. Critical temperature For T 10 -9 size of probe > 1 cm -ln(k/t) κ T c =0.115 T/t=0.05 T/t=0.1

52. Below the critical temperature : temperature in units of t antiferro- magnetic order parameter T c /t = 0.115 U = 3 Infinite-volume-correlation-length becomes larger than sample size finite sample ≈ finite k : order remains effectively

53. Pseudo-critical temperature T pc Limiting temperature at which bosonic mass term vanishes ( κ becomes nonvanishing ) It corresponds to a diverging four-fermion coupling This is the “critical temperature” computed in MFT ! Pseudo-gap behavior below this temperature

54. Pseudocritical temperature T pc μ TcTc MFT(HF) Flow eq.

55. Below the pseudocritical temperature the reign of the goldstone bosons effective nonlinear O(3) – σ - model

56. critical behavior for interval T c < T < T pc evolution as for classical Heisenberg model cf. Chakravarty,Halperin,Nelson

57. critical correlation length c,β : slowly varying functions exponential growth of correlation length compatible with observation ! at T c : correlation length reaches sample size !

58. critical behavior for order parameter and correlation function

59. Mermin-Wagner theorem ? No spontaneous symmetry breaking of continuous symmetry in d=2 ! of continuous symmetry in d=2 !

60. end