Phase transitions in Hubbard Model

1. Phase transitions in Hubbard Model

2. Anti-ferromagnetic andsuperconducting 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 ? Nospontaneous symmetry breaking of continuous symmetry in d=2 ! not valid in practice !

5. Phase diagram Pseudo-critical temperature

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

7. Flow equation for average potential

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

9. Scaling form of evolution equation 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. Tetradis …

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 For low enough k : fermion degrees of freedom decouple effectively crucial question : running of κ ( location of minimum of effective potential , renormalized , dimensionless )

12. Critical temperature For T<Tc : κ remains positive for k/t > 10-9 size of probe > 1 cm κ T/t=0.05 T/t=0.1 Tc=0.115 -ln(k/t) 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 Tc < T < Tpc 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 Tc : correlation length reaches sample size !

16. Mermin-Wagner theorem ? Nospontaneous symmetry breaking of continuous symmetry in d=2 ! not valid in practice !

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

18. Action for Hubbard model

19. Truncation for flowing action

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

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

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 k-dependent field redefinition H.Gies , … absorbs four-fermion coupling

27. running Yukawa couplings

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

29. Pseudo-critical temperature Tpc 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 Tpc MFT(HF) Flow eq. Tc μ

31. Critical temperature For T<Tc : κ remains positive for k/t > 10-9 size of probe > 1 cm κ T/t=0.05 T/t=0.1 Tc=0.115 -ln(k/t) 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. Flow equation contains correctly the non-perturbative information ! (essential scaling usually described by vortices) Essential scaling : d=2,N=2 Von Gersdorff …

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

36. Temperature dependent anomalous dimension η η T/Tc

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

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

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 order parameters can be computed as function of temperature and chemical potential competing orders further quantitative progress possible

45. changing degrees of freedom

46. flowing bosonisation • adapt bosonisation to every scale k such that is translated to bosonic interaction k-dependent field redefinition H.Gies , … absorbs four-fermion coupling

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

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

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

50. Mean field ambiguity Artefact of approximation … cured by inclusion of bosonic fluctuations J.Jaeckel,… Tc Um= Uρ= U/2 U m= U/3 ,Uρ = 0 μ mean field phase diagram