1. Unification from Functional Renormalization

2. 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

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

4. 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…

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

6. Scalar field theory Scalar field theory

7. Flow equation for average potential

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

9. Infrared cutoff

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

11. 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.

12. 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 )

13. 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

14. Critical exponents, d=3 ERGE world

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

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

17. 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 …

18. 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

19. 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

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

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

22. Unification from Functional Renormalization ☺fluctuations in d=0,1,2,3,4,... ☺linear and non-linear sigma models ☺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 homogenous systems and local disorder homogenous systems and local disorder equilibrium and out of equilibrium equilibrium and out of equilibrium

23. Exact renormalization group equation 15 years getting adult...

24. 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

25. 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 )

26. 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

27. 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 !

28. 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

29. 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”

30. Proof of exact flow equation sources j can multiply arbitrary operators φ : associated fields

31. 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

34. convergence and errors apparent fast convergence : no series resummation apparent fast convergence : no series resummation rough error estimate by different cutoffs and truncations, Fierz ambiguity etc. rough error estimate by different cutoffs and truncations, Fierz ambiguity etc. in general : understanding of physics crucial in general : understanding of physics crucial no standardized procedure no standardized procedure

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

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

37. changing degrees of freedom

38. Anti-ferromagnetic order in the Hubbard model A functional renormalization group study T.Baier, E.Bick, … C.Krahl

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

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

41. 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)

42. 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

43. 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

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

45. source term is now linear in the bosonic fields

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

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

48. 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,…

49. Bosonisation and the mean field ambiguity

50. Bosonic fluctuations fermion loops boson loops mean field theory

51. 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

52. Modification of evolution of couplings … Choose α k in order to absorb the four fermion coupling in corresponding channel Evolution with k-dependent field variables Bosonisation

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

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

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

56. 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

57. 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

58. 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

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

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

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

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

63. 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 !

64. Unification from Functional Renormalization fluctuations in d=0,1,2,3,4,... fluctuations in d=0,1,2,3,4,... ☺linear and non-linear sigma models vortices and perturbation theory vortices and perturbation theory ☺bosonic and fermionic models relativistic and non-relativistic physics relativistic and non-relativistic physics ☺classical and quantum statistics ☺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

65. non – relativistic bosons S. Floerchinger, … see also N. Dupuis arbitrary d, here d=2

66. flow of kinetic and gradient coefficients

67. density depletion T=0 Bogoliubov

68. sound velocity T=0 Bogoliubov T=0

69. critical temperature depends on size of probe T c vanishes logarithmically for infinite volume

70. condensate and superfluid density superfluid density condensate density depends on probe size

71. thermodynamics for large finite systems

72. Unification from Functional Renormalization fluctuations in d=0,1,2,3,4,... fluctuations in d=0,1,2,3,4,... 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 ☺classical and quantum statistics ☺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

73. wide applications particle physics gauge theories, QCD gauge theories, QCD Reuter,…, Marchesini et al, Ellwanger et al, Litim, Pawlowski, Gies,Freire, Morris et al., Braun, many others Reuter,…, Marchesini et al, Ellwanger et al, Litim, Pawlowski, Gies,Freire, Morris et al., Braun, 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,

74. 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, Saueressig Saueressig

75. 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, Blaizot, Benitez, Chatie, Mendes-Galain, Wschebor CW, Tetradis, Aoki, Morikawa, Souma, Sumi, Terao, Morris, Graeter, v.Gersdorff, Litim, Berges, Mouhanna, Delamotte, Canet, Bervilliers, Blaizot, Benitez, Chatie, Mendes-Galain, Wschebor Hubbard model Hubbard model Baier, Bick,…, Metzner et al, Salmhofer et al, Honerkamp et al, Krahl, Kopietz et al, Katanin, Pepin, Tsai, Strack, Baier, Bick,…, Metzner et al, Salmhofer et al, Honerkamp et al, Krahl, Kopietz et al, Katanin, Pepin, Tsai, Strack, Husemann, Lauscher Husemann, Lauscher

76. wide applications condensed matter quantum criticality quantum criticality Floerchinger, Dupuis, Sengupta, Jakubczyk, Floerchinger, Dupuis, Sengupta, Jakubczyk, sine- Gordon model sine- Gordon model Nagy, Polonyi Nagy, Polonyi disordered systems disordered systems Tissier, Tarjus, Delamotte, Canet Tissier, Tarjus, Delamotte, Canet

77. 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,…

78. wide applications condensed matter crossover phenomena crossover phenomena Bornholdt, Tetradis,… 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, Meden, Schoeller, Gasenzer, Pawlowski, Berges, Pletyukov, Reininghaus Delamotte, Tissier, Canet, Pietroni, Meden, Schoeller, Gasenzer, Pawlowski, Berges, Pletyukov, Reininghaus

79. 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 nuclear interactions nuclear interactions Schwenk Schwenk

80. wide applications ultracold atoms Feshbach resonances Feshbach resonances Diehl, Krippa, Birse, Gies, Pawlowski, Floerchinger, Scherer, Krahl, Diehl, Krippa, Birse, Gies, Pawlowski, Floerchinger, Scherer, Krahl, BEC BEC Blaizot, Wschebor, Dupuis, Sengupta, Floerchinger Blaizot, Wschebor, Dupuis, Sengupta, Floerchinger