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Emergence of new laws with Functional Renormalization

Understanding the scale-dependent laws in physics and how they transition from microscopic to effective theories is a key problem in the field. Concepts like running couplings, flowing functions, and scaling forms play a crucial role in this phenomenon. The evolution equations, critical exponents, and phase transitions described through functional renormalization offer a unified approach to diverse scalar models. By exploring the Ising model, CO2 critical exponents, and solutions of partial differential equations, this study yields significant non-perturbative insights. Examples like the Kosterlitz-Thouless phase transition showcase the essential scaling behavior captured through renormalization. Through the flow of functionals and emergence of new structures, this research contributes to the unification of abstract laws in physics, encompassing quantum gravity and grand unification theories, with a focus on critical phenomena, renormalization, and the complexity of functional theories.

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Emergence of new laws with Functional Renormalization

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  1. Emergence of new laws withFunctional Renormalization

  2. different laws at different scales fluctuations wash out many details of microscopic laws new structures as bound states or collective phenomena emerge elementary particles – earth – Universe : key problem in Physics !

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

  4. flowing action Wikipedia

  5. flowing action microscopic law macroscopic law infinitely many couplings

  6. effective theories planets fundamental microscopic law for matter in solar system: Schroedinger equation for many electrons and nucleons, in gravitational potential of sun with electromagnetic and gravitational interactions (strong and weak interactions neglected)

  7. effective theory for planets at long distances , large time scales : point-like planets , only mass of planets plays a role effective theory : Newtonian mechanics for point particles loss of memory new simple laws only a few parameters : masses of planets determined by microscopic parameters + history

  8. QCD : Short and long distance degrees of freedom are different ! Short distances : quarks and gluons Long distances : baryons and mesons How to make the transition? confinement/chiral symmetry breaking

  9. functional renormalization transition from microscopic to effective theory is made continuous effective laws depend on scale k flow in space of theories flow from simplicity to complexity – if theory is simple for large k or opposite , if theory gets simple for small k

  10. Scales in strong interactions simple complicated simple

  11. flow of functions

  12. Effective potential includes all fluctuations

  13. Scalar field theory

  14. Flow equation for average potential

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

  16. Infrared cutoff

  17. Wave function renormalization and anomalous dimension for Zk (φ,q2) : flow equation isexact !

  18. 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 …

  19. unified approach choose N choose d choose initial form of potential run ! ( quantitative results : systematic derivative expansion in second order in derivatives )

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

  21. Flow of effective potential Ising model CO2 Critical exponents Experiment : T* =304.15 K p* =73.8.bar ρ* = 0.442 g cm-2 S.Seide …

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

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

  24. Flow equation contains correctly the non-perturbative information ! (essential scaling usually described by vortices) Essential scaling : d=2,N=2 Von Gersdorff …

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

  26. Temperature dependent anomalous dimension η η T/Tc

  27. 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

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

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

  30. flow of functionals f(x) f [φ(x)]

  31. Exact renormalization group equation

  32. some history … ( the parents ) exact RG equations : Symanzik eq. , Wilson eq. , Wegner-Houghton eq. , Polchinski eq. , mathematical physics 1PI : RG for 1PI-four-point function and hierarchy Weinberg formal Legendre transform of Wilson eq. Nicoll, Chang non-perturbative flow : d=3 : sharp cutoff , no wave function renormalization or momentum dependence Hasenfratz2

  33. flow equations and composite degrees of freedom

  34. Flowing quark interactions U. Ellwanger,… Nucl.Phys.B423(1994)137

  35. Flowing four-quark vertex emergence of mesons

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

  37. changing degrees of freedom

  38. Anti-ferromagnetic order in the Hubbard model transition from microscopic theory for fermions to macroscopic theory for bosons T.Baier, E.Bick, … C.Krahl, J.Mueller, S.Friederich

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

  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 • insert identity in functional integral ( Hubbard-Stratonovich transformation ) • 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)

  42. Partially bosonised functional integral Bosonic integration is Gaussian or: solve bosonic field equation as functional of fermion fields and reinsert into action equivalent to fermionic functional integral if

  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 effective action treats fermions and composite bosons on equal footing !

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

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

  48. 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

  49. partial bosonisation and the mean field ambiguity

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