1. Emergence of new laws with Functional Renormalization

3. different laws at different scales fluctuations wash out many details of microscopic laws fluctuations wash out many details of microscopic laws new structures as bound states or collective phenomena emerge new structures as bound states or collective phenomena emerge elementary particles – earth – Universe : elementary particles – earth – Universe : key problem in Physics ! key problem in Physics !

4. scale dependent laws scale dependent ( running or flowing ) couplings scale dependent ( running or flowing ) couplings flowing functions flowing functions flowing functionals flowing functionals

5. flowing action Wikipedia

6. flowing action microscopic law macroscopic law infinitely many couplings

7. effective theories planets fundamental microscopic law for matter in solar system: fundamental microscopic law for matter in solar system: Schroedinger equation for many electrons and nucleons, Schroedinger equation for many electrons and nucleons, in gravitational potential of sun in gravitational potential of sun with electromagnetic and gravitational interactions (strong and weak interactions neglected) with electromagnetic and gravitational interactions (strong and weak interactions neglected)

8. effective theory for planets at long distances, large time scales : point-like planets, only mass of planets plays a role point-like planets, only mass of planets plays a role effective theory : Newtonian mechanics for point particles effective theory : Newtonian mechanics for point particles loss of memory loss of memory new simple laws new simple laws only a few parameters : masses of planets only a few parameters : masses of planets determined by microscopic parameters + history determined by microscopic parameters + history

9. QCD : Short and long distance degrees of freedom are different ! Short distances : quarks and gluons Short distances : quarks and gluons Long distances : baryons and mesons Long distances : baryons and mesons How to make the transition? How to make the transition? confinement/chiral symmetry breaking confinement/chiral symmetry breaking

10. functional renormalization transition from microscopic to effective theory is made continuous transition from microscopic to effective theory is made continuous effective laws depend on scale k effective laws depend on scale k flow in space of theories flow in space of theories flow from simplicity to complexity – flow from simplicity to complexity – if theory is simple for large k if theory is simple for large k or opposite, if theory gets simple for small k or opposite, if theory gets simple for small k

11. Scales in strong interactions simple complicated simple

12. flow of functions

13. Effective potential includes all fluctuations

14. Scalar field theory Scalar field theory

15. Flow equation for average potential

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

17. Infrared cutoff

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

19. 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.

20. unified approach choose N choose N choose d choose d choose initial form of potential choose initial form of potential run ! run ! ( quantitative results : systematic derivative expansion in second order in derivatives ) ( quantitative results : systematic derivative expansion in second order in derivatives )

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

22. 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

23. critical exponents, BMW approximation Blaizot, Benitez, …, Wschebor

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

25. 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 …

26. 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

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

28. Running renormalized d-wave superconducting order parameter κ in doped Hubbard (-type ) model κ - ln (k/Λ) TcTc T>T c T<T c C.Krahl,… macroscopic scale 1 cm location of minimum of u local disorder pseudo gap

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

30. unification complexity abstract laws quantum gravity grand unification standard model electro-magnetism gravity Landau universal functional theory critical physics renormalization

31. flow of functionals f(x) f [φ(x)]

32. Exact renormalization group equation

33. some history … ( the parents ) exact RG equations : exact RG equations : Symanzik eq., Wilson eq., Wegner-Houghton eq., Polchinski eq., Symanzik eq., Wilson eq., Wegner-Houghton eq., Polchinski eq., mathematical physics mathematical physics 1PI : RG for 1PI-four-point function and hierarchy 1PI : RG for 1PI-four-point function and hierarchy Weinberg Weinberg formal Legendre transform of Wilson eq. formal Legendre transform of Wilson eq. Nicoll, Chang Nicoll, Chang non-perturbative flow : non-perturbative flow : d=3 : sharp cutoff, d=3 : sharp cutoff, no wave function renormalization or momentum dependence no wave function renormalization or momentum dependence Hasenfratz 2 Hasenfratz 2

34. flow equations and composite degrees of freedom

35. Flowing quark interactions U. Ellwanger,… Nucl.Phys.B423(1994)137

36. Flowing four-quark vertex emergence of mesons

37. Floerchinger, Scherer, Diehl,… see also Diehl, Floerchinger, Gies, Pawlowski,… BCSBEC free bosons interacting bosons BCS Gorkov BCS – BEC crossover

38. changing degrees of freedom

39. Anti-ferromagnetic order in the Hubbard model transition from microscopic theory for fermions to macroscopic theory for bosons T.Baier, E.Bick, … C.Krahl, J.Mueller, S.Friederich

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

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

42. 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)

43. 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

44. more bosons … additional fields may be added formally : only mass term + source term : decoupled boson introduction of boson fields not linked to Hubbard-Stratonovich transformation

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

46. source term is now linear in the bosonic fields effective action treats fermions and composite bosons on equal footing !

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

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

49. 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,…

50. partial bosonisation and the mean field ambiguity

51. Bosonic fluctuations fermion loops boson loops mean field theory

52. flowing bosonisation adapt bosonisation to every scale k such that adapt bosonisation 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

53. flowing bosonisation Choose α k in order to absorb the four fermion coupling in corresponding channel Evolution with k-dependent field variables modified flow of couplings

54. Bosonisation cures mean field ambiguity TcTc U ρ /t MFT Flow eq. HF/SD

55. Flow equation for the Hubbard model T.Baier, E.Bick, …,C.Krahl, J.Mueller, S.Friederich

56. 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

57. 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 local disorder pseudo gap SSB

58. Mermin-Wagner theorem ? No spontaneous symmetry breaking of continuous symmetry in d=2 ! of continuous symmetry in d=2 ! not valid in practice ! not valid in practice !

59. 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

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

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

62. Critical temperature -ln(k/t) κ T c =0.115 T/t=0.05 T/t=0.1 local disorder pseudo gap SSB only Goldstone bosons matter !

63. critical behavior for interval T c < T < T pc evolution as for classical Heisenberg model cf. Chakravarty,Halperin,Nelson dimensionless coupling of non-linear sigma-model : g 2 ~ κ -1 two-loop beta function for g

64. effective theory non-linear O(3)-sigma-model asymptotic freedom from fermionic microscopic law to bosonic macroscopic law

65. transition to linear sigma-model large coupling regime of non-linear sigma-model : small renormalized order parameter κ transition to symmetric phase again change of effective laws : linear sigma-model is simple, strongly coupled non-linear sigma-model is complicated

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

67. conclusion functional renormalization offers an efficient method for adding new relevant degrees of freedom or removing irrelevant degrees of freedom continuous description of the emergence of new laws

68. Unification from Functional Renormalization fluctuations in d=0,1,2,3,... fluctuations in d=0,1,2,3,... linear and non-linear sigma models linear and non-linear sigma models vortices and perturbation theory vortices and perturbation theory bosonic and fermionic models bosonic and fermionic models relativistic and non-relativistic physics relativistic and non-relativistic physics classical and quantum statistics classical and quantum statistics non-universal and universal aspects non-universal and universal aspects homogenous systems and local disorder homogenous systems and local disorder equilibrium and out of equilibrium equilibrium and out of equilibrium

69. end

70. unification: functional integral / flow equation simplicity of average action simplicity of average action explicit presence of scale explicit presence of scale differentiating is easier than integrating… differentiating is easier than integrating…

71. qualitative changes that make non- perturbative physics accessible : ( 1 ) basic object is simple ( 1 ) basic object is simple average action ~ classical action average action ~ classical action ~ generalized Landau theory ~ generalized Landau theory direct connection to thermodynamics direct connection to thermodynamics (coarse grained free energy ) (coarse grained free energy )

72. qualitative changes that make non- perturbative physics accessible : ( 2 ) Infrared scale k instead of Ultraviolet cutoff Λ instead of Ultraviolet cutoff Λ short distance memory not lost no modes are integrated out, but only part of the fluctuations is included simple one-loop form of flow simple comparison with perturbation theory

73. infrared cutoff k cutoff on momentum resolution or frequency resolution or frequency resolution e.g. distance from pure anti-ferromagnetic momentum or from Fermi surface e.g. distance from pure anti-ferromagnetic momentum or from Fermi surface intuitive interpretation of k by association with physical IR-cutoff, i.e. finite size of system : arbitrarily small momentum differences cannot be resolved ! arbitrarily small momentum differences cannot be resolved !

74. qualitative changes that make non- perturbative physics accessible : ( 3 ) only physics in small momentum range around k matters for the flow ERGE regularization simple implementation on lattice artificial non-analyticities can be avoided

75. qualitative changes that make non- perturbative physics accessible : ( 4 ) flexibility ( 4 ) flexibility change of fields microscopic or composite variables simple description of collective degrees of freedom and bound states many possible choices of “cutoffs”