1. Challenges for Functional Renormalization

2. Exact renormalization group equation

3. important success in many areas first rate tool if perturbative expansions or numerical simulations fail or are difficult models with fermions models with fermions gravity gravity models with largely different length scales models with largely different length scales non-perturbative renormalizability non-perturbative renormalizability

4. 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 key problem in Physics ! key problem in Physics !

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

6. flowing action Wikipedia

7. flowing action microscopic law macroscopic law infinitely many couplings

8. 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 if theory is simple for large k flow from simplicity to complexity 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

9. challenges for functional renormalization reliability and error estimates reliability and error estimates accessibility accessibility exploration of new terrain exploration of new terrain precision and benchmarking precision and benchmarking

10. truncation error

11. structure of truncated flow equation g : flowing data ς: flow generators

12. flowing data typically, g can be viewed as functions typically, g can be viewed as functions effective potential U( ρ ) inverse relativistic propagator P(p 2 ) inverse non-relativistic propagator P( ω, p 2 ) momentum dependent four-fermion vertex

13. truncation finite number of functions functions parameterized by finite set of data e.g. e.g. polynomial expansion polynomial expansion function values at given arguments function values at given arguments ( can be single coupling) ( can be single coupling) truncation : limitation to restricted set of data (finite set for numerical purposes)

14. exact flow equations for given set of data : for given set of data : flow generators ς can be computed exactly as formal expressions

15. O(N) – scalar model first order derivative expansion flowing data g : U( ρ ), Z( ρ ), Y( ρ ) exact generator for U:

16. momentum integration one loop form of exact flow equation : σ (q) : input functions one d-dimensional momentum integration necessary, ( sometimes analytical integration possible )

17. specification For finite set of data the system of flow equations is not closed ! σ (q) cannot be computed uniquely from g one needs prescription how σ (q) is determined in terms of g

18. specification parameters specification typically involves specification parameters s

19. Lowest order derivative expansion for scalar O(N) model Z( ρ )= Z Z( ρ )= Z Y( ρ )=0 Y( ρ )=0 flowing data : U( ρ ), Z flowing data : U( ρ ), Z

20. Flow equation for average potential

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

22. Scalar field theory Scalar field theory

23. specification (1) one has to specify the exact definition of Z from inverse propagator P(q) inverse propagator P(q) P: second functional derivative at minimum of effective potential - Goldstone mode or radial mode Z= ∂ P/ ∂ q 2 at q 2 =0 or Z= ∂ P/ ∂ q 2 at q 2 =0 or Z=(P(q 2 =ck 2 )-P(0))/ck 2 Z=(P(q 2 =ck 2 )-P(0))/ck 2 c is one of the specification parameters c is one of the specification parameters

24. Which forms of effective action are compatible with given data g ?

25. specification (2) different forms of inverse propagator are compatible with a given definition of Z

26. choice of inverse propagator

27. physics knowledge can be put into choice of general form of input functions !

28. flow parameters w specification parameters specification parameters cutoff parameters cutoff parameters bosonization parameters bosonization parameters

29. error estimate vary w within certain priors vary w within certain priors

30. accessibility

31. public program with structure for first step : individual routines from users / library

32. numerical momentum integration

33. update of flow

34. precision and benchmarking

35. benchmarks (1) universal critical physics: universal critical physics: O(N)-models : exponents, amplitude ratios, equation of state ( including non-perturbative physics as Kosterlitz -Thouless transition ) O(N)-models : exponents, amplitude ratios, equation of state ( including non-perturbative physics as Kosterlitz -Thouless transition ) non-abelian non-linear sigma-models in d=2 : mass gap, correlation functions non-abelian non-linear sigma-models in d=2 : mass gap, correlation functions Ising model on lattice and other exactly solvable models Ising model on lattice and other exactly solvable models

36. benchmarks (2) quantum mechanics quantum mechanics a) as d=1 functional integral a) as d=1 functional integral b) limit n → 0, T → 0 of many body system, b) limit n → 0, T → 0 of many body system, d=3+1 d=3+1 scattering length for atoms, dimers, Efimov effect with Efimov parameter scattering length for atoms, dimers, Efimov effect with Efimov parameter BCS-BEC crossover for ultracold atoms : BCS-BEC crossover for ultracold atoms : all T, n, a all T, n, a

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

38. explore new terrain up to you ! up to you !

39. Phase transitions in Hubbard Model

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

41. Phase diagram AF SC

42. 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 !

43. Phase diagram Pseudo-critical temperature

44. challenges for functional renormalization reliability and error estimates reliability and error estimates accessibility accessibility exploration of new terrain exploration of new terrain precision and benchmarking precision and benchmarking

45. conclusions functional renormalization : functional renormalization : bright future bright future substantial work ahead substantial work ahead

46. end