1. Phase transitions in Hubbard Model

2. Anti-ferromagnetic and superconducting order in the Hubbard model A functional renormalization group study T.Baier, E.Bick, … C.Krahl, J.Mueller, S.Friederich

3. Phase diagram AF SC

4. 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 !

5. Phase diagram Pseudo-critical temperature

6. Goldstone boson fluctuations spin waves ( anti-ferromagnetism ) spin waves ( anti-ferromagnetism ) electron pairs ( superconductivity ) electron pairs ( superconductivity )

7. Flow equation for average potential

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

9. 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.

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

11. Anti-ferromagnetism in Hubbard model SO(3) – symmetric scalar model coupled to fermions SO(3) – symmetric scalar model coupled to fermions For low enough k : fermion degrees of freedom decouple effectively For low enough k : fermion degrees of freedom decouple effectively crucial question : running of κ ( location of minimum of effective potential, renormalized, dimensionless ) crucial question : running of κ ( location of minimum of effective potential, renormalized, dimensionless )

12. 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

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

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

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

16. 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 !

17. 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

18. Action for Hubbard model

19. Truncation for flowing action

20. Additional bosonic fields anti-ferromagnetic anti-ferromagnetic charge density wave charge density wave s-wave superconducting s-wave superconducting d-wave superconducting d-wave superconducting initial values for flow : bosons are decoupled auxiliary fields ( microscopic action ) initial values for flow : bosons are decoupled auxiliary fields ( microscopic action )

21. Effective potential for bosons SYM SSB microscopic : only “mass terms”

22. Yukawa coupling between fermions and bosons Microscopic Yukawa couplings vanish !

23. Kinetic terms for bosonic fields anti-ferromagnetic boson d-wave superconducting boson

24. incommensurate anti-ferromagnetism commensurate regime : incommensurate regime :

25. infrared cutoff linear cutoff ( Litim )

26. flowing bosonisation effective four-fermion coupling in appropriate channel is translated to bosonic interaction at every scale k H.Gies, … k-dependent field redefinition absorbs four-fermion coupling

27. running Yukawa couplings

28. flowing boson mass terms SYM : close to phase transition

29. 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

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

31. 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

32. Phase diagram Pseudo-critical temperature

33. spontaneous symmetry breaking of abelian continuous symmetry in d=2 Bose –Einstein condensate Superconductivity in Hubbard model Kosterlitz – Thouless phase transition

34. 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 …

35. 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

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

37. 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

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

39. order parameters in Hubbard model

40. Competing orders AF SC

41. Anti-ferromagnetism suppresses superconductivity

42. coexistence of different orders ?

43. quartic couplings for bosons

44. conclusions functional renormalization gives access to low temperature phases of Hubbard model functional renormalization gives access to low temperature phases of Hubbard model order parameters can be computed as function of temperature and chemical potential order parameters can be computed as function of temperature and chemical potential competing orders competing orders further quantitative progress possible further quantitative progress possible

45. changing degrees of freedom

46. 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

47. 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

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

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

50. 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,…

51. Bosonisation and the mean field ambiguity

52. Bosonic fluctuations fermion loops boson loops mean field theory

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

54. end

55. quartic couplings for bosons

56. kinetic and gradient terms for bosons

57. fermionic wave function renormalization