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Unification from Functional Renormalization

Unification from Functional Renormalization . Unification from Functional Renormalization . fluctuations in d=0,1,2,3,... linear and non-linear sigma models vortices and perturbation theory bosonic and fermionic models relativistic and non-relativistic physics

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Unification from Functional Renormalization

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  1. Unification fromFunctional Renormalization

  2. Unification fromFunctional Renormalization • fluctuations in d=0,1,2,3,... • linear and non-linear sigma models • vortices and perturbation theory • bosonic and fermionic models • relativistic and non-relativistic physics • classical and quantum statistics • non-universal and universal aspects • homogenous systems and local disorder • equilibrium and out of equilibrium

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

  4. unification:functional integral / flow equation • simplicity of average action • explicit presence of scale • differentiating is easier than integrating…

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

  6. Scalar field theory

  7. Flow equation for average potential

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

  9. Infrared cutoff

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

  11. 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 …

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

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

  14. Critical exponents , d=3 ERGE world ERGE world

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

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

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

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

  19. 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

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

  21. Temperature dependent anomalous dimension η η T/Tc

  22. Unification fromFunctional Renormalization ☺fluctuations in d=0,1,2,3,4,... ☺linear and non-linear sigma models ☺vortices and perturbation theory • bosonic and fermionic models • relativistic and non-relativistic physics • classical and quantum statistics ☺non-universal and universal aspects • homogenous systems and local disorder • equilibrium and out of equilibrium

  23. Exact renormalization group equation 15 years getting adult...

  24. 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

  25. qualitative changes that make non-perturbative physics accessible : ( 1 ) basic object is simple average action ~ classical action ~ generalized Landau theory direct connection to thermodynamics (coarse grained free energy )

  26. qualitative changes that make non-perturbative physics accessible : ( 2 ) Infrared scale k 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

  27. infrared cutoff k cutoff on momentum resolution or frequency resolution 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 !

  28. 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

  29. qualitative changes that make non-perturbative physics accessible : ( 4 ) flexibility change of fields microscopic or composite variables simple description of collective degrees of freedom and bound states many possible choices of “cutoffs”

  30. Proof of exact flow equation sources j can multiply arbitrary operators φ : associated fields

  31. Truncations Functional differential equation – cannot be solved exactly Approximative solution by truncationof most general form of effective action

  32. convergence and errors • apparent fast convergence : no series resummation • rough error estimate by different cutoffs and truncations , Fierz ambiguity etc. • in general : understanding of physics crucial • no standardized procedure

  33. including fermions : no particular problem !

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

  35. changing degrees of freedom

  36. Anti-ferromagnetic order in the Hubbard model A functional renormalization group study T.Baier, E.Bick, … C.Krahl

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

  38. Fermion bilinears Introduce sources for bilinears Functional variation with respect to sources J yields expectation values and correlation functions

  39. 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)

  40. 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

  41. 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

  42. fermion – boson action fermion kinetic term boson quadratic term (“classical propagator” ) Yukawa coupling

  43. source term is now linear in the bosonic fields

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

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

  46. 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

  47. Bosonisation and the mean field ambiguity

  48. Bosonic fluctuations boson loops fermion loops mean field theory

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