1. Flow equations and phase transitions

3. phase transitions

4. QCD – phase transition Quark –gluon plasma Quark –gluon plasma Gluons : 8 x 2 = 16 Gluons : 8 x 2 = 16 Quarks : 9 x 7/2 =12.5 Quarks : 9 x 7/2 =12.5 Dof : 28.5 Dof : 28.5 Chiral symmetry Chiral symmetry Hadron gas Light mesons : 8 (pions : 3 ) Dof : 8 Chiral sym. broken Large difference in number of degrees of freedom ! Strong increase of density and energy density at T c !

5. Understanding the phase diagram

6. quark-gluon plasma “deconfinement” quark matter : superfluid B spontaneously broken nuclear matter : B,isospin (I 3 ) spontaneously broken, S conserved Phase diagram for m s > m u,d vacuum

7. Order parameters Nuclear matter and quark matter are separated from other phases by true critical lines Nuclear matter and quark matter are separated from other phases by true critical lines Different realizations of global symmetries Different realizations of global symmetries Quark matter: SSB of baryon number B Quark matter: SSB of baryon number B Nuclear matter: SSB of combination of B and isospin I 3 Nuclear matter: SSB of combination of B and isospin I 3 neutron-neutron condensate neutron-neutron condensate

8. quark-gluon plasma “deconfinement” quark matter : superfluid B spontaneously broken nuclear matter : superfluid B, isospin (I 3 ) spontaneously broken, S conserved vacuum Phase diagram for m s > m u,d

9. deconfinement liquid vacuum

10. nuclear matter B,isospin (I 3 ) spontaneously broken, S conserved quark matter : superfluid B spontaneously broken quark-gluon plasma “deconfinement” Phase diagram for m s > m u,d

11. Phase diagram = σ ≠ 0 = σ ≠ 0 ≈0 ≈0 quark-gluon plasma “deconfinement” vacuum quark matter, nuclear mater: superfluid B spontaneously broken

12. First order phase transition line First order phase transition line hadrons hadrons quarks and gluons μ=923MeV transition to nuclear nuclear matter matter

13. Exclusion argument hadronic phase with sufficient production of Ω : excluded !! excluded !! P.Braun-Munzinger, J.Stachel,… T c ≈ T ch

14. “minimal” phase diagram for equal nonzero quark masses “minimal” phase diagram for equal nonzero quark masses

15. strong interactions in high T phase

17. presence of strong interactions: crossover or first order phase transition if no global order parameter distinguishes phases if no global order parameter distinguishes phases ( second order only at critical endpoint ) ( second order only at critical endpoint ) use flow non-perturbative equations

18. Electroweak phase transition ? 10 -12 s after big bang 10 -12 s after big bang fermions and W-,Z-bosons get mass fermions and W-,Z-bosons get mass standard model : crossover standard model : crossover baryogenesis if first order baryogenesis if first order ( only for some SUSY – models ) ( only for some SUSY – models ) bubble formation of “ our vacuum “ bubble formation of “ our vacuum “ Reuter,Wetterich ‘93 Kuzmin,Rubakov,Shaposhnikov ‘85, Shaposhnikov ‘87

19. strong electroweak interactions responsible for crossover

20. Electroweak phase diagram M.Reuter,C.WetterichNucl.Phys.B408,91(1993)

21. Masses of excitations (d=3) O.Philipsen,M.Teper,H.Wittig ‘97 small M H large M H

22. Continuity

23. BEC – BCS crossover Bound molecules of two atoms Bound molecules of two atoms on microscopic scale: on microscopic scale: Bose-Einstein condensate (BEC ) for low T Bose-Einstein condensate (BEC ) for low T Fermions with attractive interactions Fermions with attractive interactions (molecules play no role ) : (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

24. scattering length BCSBEC

25. 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

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

27. Flow equations

28. 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

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

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

31. Scalar field theory Scalar field theory

32. Flow equation for average potential

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

34. Infrared cutoff

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

36. 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 )

37. 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

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

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

40. 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 …

41. 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

42. Exact renormalization group equation

43. 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,

44. 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

45. 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

46. 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

47. 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,…

48. 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

49. 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

50. 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

51. conclusions There is still a lot of interesting theory to learn for an understanding of phase transitions There is still a lot of interesting theory to learn for an understanding of phase transitions Perhaps non-perturbative low equations can help Perhaps non-perturbative low equations can help

53. 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 )

54. 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

55. 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 !

56. 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

57. 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”

58. some history … 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