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Unification from Functional Renormalization

Unification from Functional Renormalization. Wegner, Houghton. /. Effective potential includes all fluctuations. 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

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Unification from Functional Renormalization

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  1. Unification fromFunctional Renormalization

  2. Wegner, Houghton /

  3. Effective potential includes all fluctuations

  4. 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

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

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

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

  8. Scalar field theory

  9. Flow equation for average potential

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

  11. Infrared cutoff

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

  13. 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 …

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

  15. 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 …

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

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

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

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

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

  21. 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

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

  23. Temperature dependent anomalous dimension η η T/Tc

  24. 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

  25. Exact renormalization group equation

  26. 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

  27. 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 )

  28. 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

  29. 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 !

  30. 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

  31. 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”

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

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

  34. 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

  35. including fermions : no particular problem !

  36. Universality in ultra-cold fermionic atom gases

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

  38. BEC – BCS crossover Bound molecules of two atoms on microscopic scale: Bose-Einstein condensate (BEC ) for low T Fermions with attractive interactions (molecules play no role ) : BCS – superfluidity at low T by condensation of Cooper pairs Crossoverby Feshbach resonance as a transition in terms of external magnetic field

  39. Feshbach resonance H.Stoof

  40. scattering length BEC BCS

  41. chemical potential BCS BEC inverse scattering length

  42. concentration • c = a kF , a(B) : scattering length • needs computation of density n=kF3/(3π2) dilute dense dilute non- interacting Fermi gas non- interacting Bose gas T = 0 BCS BEC

  43. universality same curve for Li and K atoms ? dilute dense dilute T = 0 BCS BEC

  44. different methods Quantum Monte Carlo

  45. who cares about details ? a theorists game …? MFT RG

  46. precision many body theory- quantum field theory - so far : • particle physics : perturbative calculations magnetic moment of electron : g/2 = 1.001 159 652 180 85 ( 76 ) ( Gabrielse et al. ) • statistical physics : universal critical exponents for second order phase transitions : ν = 0.6308 (10) renormalization group • lattice simulations for bosonic systems in particle and statistical physics ( e.g. QCD )

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

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