Functional renormalization concepts and prospects

Transcription

1 Functional renormalization concepts and prospects

2 physics at different length scales microscopic theories : where the laws are formulated effective theories : where observations are made effective theory may involve different degrees of freedom as compared to microscopic theory example: the motion of the earth around the sun does not need an understanding of nuclear burning in the sun

3 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 chiral symmetry breaking

4 collective degrees of freedom

5 Hubbard model Electrons on a cubic lattice here : on planes ( d = 2 ) Repulsive local interaction if two electrons are on the same site Hopping interaction between two neighboring sites

6 In solid state physics : model for everything Antiferromagnetism High T c superconductivity Metal-insulator transition Ferromagnetism

7 Antiferromagnetism in d=2 Hubbard model U/t = 3 antiferromagnetic order parameter T c /t = T.Baier, E.Bick, temperature in units of t

8 Collective degrees of freedom are crucial! for T < T c nonvanishing order parameter gap for fermions low energy excitations: antiferromagnetic spin waves

9 effective theory / microscopic theory sometimes only distinguished by different values of couplings sometimes different degrees of freedom

10 Functional Renormalization Group describes flow of effective action from small to large length scales perturbative renormalization : case where only couplings change, and couplings are small

11 How to come from quarks and gluons to baryons and mesons? How to come from electrons to spin waves? Find effective description where relevant degrees of freedom depend on momentum scale or resolution in space. Microscope with variable resolution: High resolution, small piece of volume: quarks and gluons Low resolution, large volume : hadrons

12

13 Wegner, Houghton /

14 effective average action

15 Effective average potential : Unified picture for scalar field theories with symmetry O(N) in arbitrary dimension d and arbitrary N linear or nonlinear sigma-model model for chiral symmetry breaking in QCD or: scalar model for antiferromagnetic spin waves (linear O(3) model ) fermions will be added later

16 Effective potential includes all fluctuations

17 Scalar field theory

18 Flow equation for average potential

19 Simple one loop structure nevertheless (almost) exact

20 Infrared cutoff

21 Partial differential equation for function U(k,φ) ) depending on two ( or more ) variables Z k = c k-η

22 Regularisation For suitable R k : Momentum integral is ultraviolet and infrared finite Numerical integration possible Flow equation defines a regularization scheme ( ERGE regularization )

23 Integration by momentum shells Momentum integral is dominated by q 2 ~ k 2. Flow only sensitive to physics at scale k

24 Wave function renormalization and anomalous dimension for Z k (φ,q 2 ) : flow equation is exact!

25 Flow of effective potential Ising model CO 2 Critical exponents Experiment : S.Seide T * = K p * =73.8.bar = g cm-2 ρ *

26 Critical exponents, d=3

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

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

29 Kosterlitz-Thouless phase transition (d=2,n=2) Correct description of phase with Goldstone boson ( infinite correlation length ) for T<T c

30 Running renormalized d-wave d superconducting order parameter κ in Hubbard model κ T>T c T c T<T c C.Krahl, -ln(k/λ) macroscopic scale 1 cm

31 Renormalized order parameter κ and gap in electron propagator Δ 100 Δ / t κ jump T/T c

32 Temperature dependent anomalous dimension η η T/T c

33 Effective average action and exact renormalization group equation

34 Generating functional

35 Effective average action Loop expansion : perturbation theory with infrared cutoff in propagator

36 Quantum effective action

37 Exact renormalization group equation

38 Proof of exact flow equation

39 Truncations Functional differential equation cannot be solved exactly Approximative solution by truncation of most general form of effective action

40

41

42 Exact flow equation for effective potential Evaluate exact flow equation for homogeneous field φ. R.h.s.. involves exact propagator in homogeneous background field φ.

43 Flow equation for average potential

44 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 chiral symmetry breaking

45 Nambu Jona-Lasinio model and more general quark meson models

46 Chiral condensate (N( f =2) Berges, Jungnickel,

47 Critical temperature, N f = 2 J.Berges,D.Jungnickel, Lattice simulation

48 temperature dependent masses pion mass sigma mass

49 Critical equation of state

50 Scaling form of equation of state Berges, Tetradis,

51 Universal critical equation of state is valid near critical temperature if the only light degrees of freedom are pions + sigma with O(4) symmetry. Not necessarily valid in QCD, even for two flavors!

52 conclusions Flow equation for effective average action: Does it work? Why does it work? When does it work? How accurately does it work?

53 Functional renormalization group for the effective average action

54 particle physics wide applications gauge theories, QCD Reuter,, Marchesini et al, Ellwanger et al, Litim, Pawlowski, Gies,Freire, Morris et al., many others electroweak interactions, gauge hierarchy problem Jaeckel,Gies, electroweak phase transition Reuter,Tetradis, Bergerhoff Bergerhoff,

55 wide applications gravity asymptotic safety Reuter, Lauscher, Schwindt et al, Percacci et al, Litim,, Fischer

56 wide applications condensed matter unified description for classical bosons CW, Tetradis, Aoki, Morikawa,, Souma, Sumi, Terao, Morris, Graeter, v.gersdorff, Litim, Berges, Mouhanna, Delamotte, Canet, Bervilliers, Hubbard model Baier,Bick Honerkamp et al, Krahl, Baier,Bick,, Metzner et al, Salmhofer et al, disordered systems Tissier, Tarjus,Delamotte, Canet

57 condensed matter wide applications equation of state for CO 2 Seide, liquid He 4 Gollisch, and He 3 Kindermann, frustrated magnets Delamotte, Mouhanna, Tissier nucleation and first order phase transitions Tetradis, Strumia,, Berges,

58 wide applications condensed matter crossover phenomena Bornholdt, Tetradis, superconductivity ( scalar QED 3 ) Bergerhoff,, Lola, Litim, Freire, non equilibrium systems Delamotte, Tissier, Canet, Pietroni

59 wide applications nuclear physics effective NJL- type models Ellwanger, Jungnickel, Berges, Tetradis,, Pirner,, Schaefer, Wambach, Kunihiro, Schwenk, di-neutron neutron condensates Birse, Krippa, equation of state for nuclear matter Berges, Jungnickel, Birse, Krippa

60 wide applications ultracold atoms Feshbach resonances Diehl, Gies, Pawlowski,, Krippa, BEC Blaizot Blaizot, Wschebor,, Dupuis, Sengupta

61 unified description of scalar models for all d and N

62 Flow equation for average potential

63 Scalar field theory

64 Simple one loop structure nevertheless (almost) exact

65 Infrared cutoff

66 Wave function renormalization and anomalous dimension for Z k (φ,q 2 ) : flow equation is exact!

67 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

68 unified approach choose N choose d choose initial form of potential run!

69 Flow of effective potential Ising model CO 2 Critical exponents Experiment : S.Seide T * = K p * =73.8.bar = g cm-2 ρ *

70 Critical exponents, d=3 ERGE world ERGE world

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

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

73 Kosterlitz-Thouless phase transition (d=2,n=2) Correct description of phase with Goldstone boson ( infinite correlation length ) for T<T c

74 Running renormalized d-wave d superconducting order parameter κ in doped Hubbard model κ T<T c T c T>T c C.Krahl, -ln(k/λ) macroscopic scale 1 cm

75 Renormalized order parameter κ and gap in electron propagator Δ in doped Hubbard model 100 Δ / t κ jump T/T c

76 Temperature dependent anomalous dimension η η T/T c

77 convergence and errors for precise results: systematic derivative expansion in second order in derivatives includes field dependent wave function renormalization Z(ρ) fourth order : similar results apparent fast convergence : no series resummation rough error estimate by different cutoffs and truncations

78 including fermions : no particular problem!

79 changing degrees of freedom

80 Antiferromagnetic order in the Hubbard model A functional renormalization group study T.Baier, E.Bick,

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

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

83 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)

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

85 fermion boson action fermion kinetic term boson quadratic term ( classical propagator ) Yukawa coupling

86 source term is now linear in the bosonic fields

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

88 Effective potential in mean field theory

89 Mean field phase diagram for two different choices of couplings same U! T c T c μ μ

90 Mean field ambiguity T c U m = U ρ = U/2 Artefact of approximation U m = U/3,U ρ = 0 μ cured by inclusion of bosonic fluctuations J.Jaeckel, mean field phase diagram

91 Rebosonization and the mean field ambiguity

92 Bosonic fluctuations fermion loops boson loops mean field theory

93 Rebosonization adapt bosonization to every scale k such that is translated to bosonic interaction H.Gies, k-dependent field redefinition absorbs four-fermion coupling

94 Modification of evolution of couplings Evolution with k-dependent field variables Rebosonisation Choose α k such that no four fermion coupling is generated

95 cures mean field ambiguity T c MFT HF/SD Flow eq. U ρ /t

96 Flow equation for the Hubbard model T.Baier, E.Bick,,C.Krahl

97 Truncation Concentrate on antiferromagnetism Potential U depends only on α = a 2

98 scale evolution of effective potential for antiferromagnetic order parameter boson contribution fermion contribution effective masses depend on α! gap for fermions ~α

99 running couplings

100 Running mass term unrenormalized mass term -ln(k/t) four-fermion interaction ~ m -2 diverges

101 dimensionless quantities renormalized antiferromagnetic order parameter κ

102 evolution of potential minimum κ 10-2 λ U/t = 3, T/t = ln(k/t)

103 Critical temperature For T<T c : κ remains positive for k/t > 10-9 size of probe > 1 cm κ T/t=0.05 T/t=0.1 -ln(k/t) T c =0.115

104 Below the critical temperature : Infinite-volume-correlation-length becomes larger than sample size finite sample finite k : order remains effectively U = 3 antiferromagnetic order parameter T c /t = temperature in units of t

105 Pseudo-critical temperature Tpc Limiting temperature at which bosonic mass term vanishes ( κ becomes nonvanishing ) It corresponds to a diverging four-fermion fermion coupling This is the critical temperature computed in MFT! Pseudo-gap behavior below this temperature

106 Pseudocritical temperature T pc MFT(HF) Flow eq. T c μ

107 Below the pseudocritical temperature the reign of the goldstone bosons effective nonlinear O(3) σ -model

108 critical behavior for interval T c < T < T pc evolution as for classical Heisenberg model cf. Chakravarty,Halperin,Nelson

109 critical correlation length c,β : slowly varying functions exponential growth of correlation length compatible with observation! at T c : correlation length reaches sample size!

110 critical behavior for order parameter and correlation function

111 Mermin-Wagner theorem? No spontaneous symmetry breaking of continuous symmetry in d=2!

112 Universality in ultra-cold fermionic atom gases

113 Universality in ultra-cold fermionic atom gases with S. Diehl, H.Gies, J.Pawlowski

114 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 Crossover by by Feshbach resonance as a transition in terms of external magnetic field

115 Feshbach resonance H.Stoof

116 scattering length BEC BCS

117 chemical potential BCS inverse scattering length BEC

118 BEC BCS crossover qualitative and partially quantitative theoretical understanding mean field theory (MFT ) and first attempts beyond T = 0 concentration : c = a k F reduced chemical potential : σ = μ/ε F Fermi momemtum : k F Fermi energy : ε F binding energy : BCS BEC

119 c = a k F concentration, a(b) : scattering length needs computation of density n=k F3 /(3π 2 ) dilute dense dilute noninteracting Fermi gas T = 0 noninteracting Bose gas BCS BEC

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

121 different methods Quantum Monte Carlo

122 who cares about details? a theorists game? RG MFT

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

124 QFT with fermions needed: universal theoretical tools for complex fermionic systems wide applications : electrons in solids, nuclear matter in neutron stars,.

125 QFT for non-relativistic fermions functional integral, action perturbation theory: Feynman rules τ : euclidean time on torus with circumference 1/T σ : effective chemical potential

126 variables ψ : Grassmann variables φ : bosonic field with atom number two What is φ? microscopic molecule, macroscopic Cooper pair? All!

127 parameters detuning ν(b) Yukawa or Feshbach coupling h φ

128 fermionic action equivalent fermionic action, in general not local

129 scattering length a a= M λ/4π broad resonance : pointlike limit large Feshbach coupling

130 parameters Yukawa or Feshbach coupling h φ scattering length a broad resonance : h φ drops out

131 concentration c

132

133 universality Are these parameters enough for a quantitatively precise description? Have Li and K the same crossover when described with these parameters? Long distance physics looses memory of detailed microscopic properties of atoms and molecules! universality for c -1 = 0 : Ho, ( ( valid for broad resonance) here: whole crossover range

134 analogy with particle physics microscopic theory not known - nevertheless macroscopic theory characterized by a finite number of renormalizable couplings m e, α ; g w, g s, M w, here : c, h φ ( only c for broad resonance )

135 analogy with universal critical exponents only one relevant parameter : T - T c

136 units and dimensions c = 1 ; ħ =1 ; k = 1 momentum ~ length -1 ~ mass ~ ev energies : 2ME ~ (momentum) 2 ( M : atom mass ) typical momentum unit : Fermi momentum typical energy and temperature unit : Fermi energy time ~ (momentum) -2 canonical dimensions different from relativistic QFT!

137 rescaled action M drops out all quantities in units of k F if

138 what is to be computed? Inclusion of fluctuation effects via functional integral leads to effective action. This contains all relevant information for arbitrary T and n!

139 effective action integrate out all quantum and thermal fluctuations quantum effective action generates full propagators and vertices richer structure than classical action

140 effective action includes all quantum and thermal fluctuations formulated here in terms of renormalized fields involves renormalized couplings

141 effective potential minimum determines order parameter condensate fraction Ω c = 2 ρ 0 /n

142 effective potential value of φ at potential minimum : order parameter, determines condensate fraction second derivative of U with respect to φ yields correlation length derivative with respect to σ yields density fourth derivative of U with respect to φ yields molecular scattering length

143 renormalized fields and couplings

144 challenge for ultra-cold atoms : Non-relativistic fermion systems with precision similar to particle physics! ( QCD with quarks )

145 results from functional renormalization group

146 physics at different length scales microscopic theories : where the laws are formulated effective theories : where observations are made effective theory may involve different degrees of freedom as compared to microscopic theory example: microscopic theory only for fermionic atoms, macroscopic theory involves bosonic collective degrees of freedom ( φ )

147 gap parameter Δ BCS for gap T = 0 BCS BEC

148 limits BCS for gap condensate fraction for bosons with scattering length 0.9 a

149 temperature dependence of condensate second order phase transition

150 condensate fraction : second order phase transition c -1 =1 free BEC c -1 =0 universal critical behavior T/T c

151 crossover phase diagram

152 shift of BEC critical temperature

153 running couplings : crucial for universality for large Yukawa couplings h φ : only one relevant parameter c all other couplings are strongly attracted to partial fixed points macroscopic quantities can be predicted in terms of c and T/ε F ( in suitable range for c -1 )

154 Flow of Yukawa coupling T=0.5, c=1 k 2 k 2

155 universality for broad resonances for large Yukawa couplings h φ : only one relevant parameter c all other couplings are strongly attracted to partial fixed points macroscopic quantities can be predicted in terms of c and T/ε F ( in suitable range for c -1 ; density sets scale )

156 universality for narrow resonances Yukawa coupling becomes additional parameter ( marginal coupling ) also background scattering important

157 Flow of Yukawa and four fermion coupling λ ψ /8π (A ) broad Feshbach resonance (C) narrow Feshbach resonance h 2 /32π

158 Universality is due to fixed points!

159 not all quantities are universal!

160 bare molecule fraction (fraction of microscopic closed channel molecules ) not all quantities are universal bare molecule fraction involves wave function renormalization that depends on value of Yukawa coupling 6 Li Experimental points by Partridge et al. B[G]

161 conclusions the challenge of precision : substantial theoretical progress needed phenomenology has to identify quantities that are accessible to precision both for experiment and theory dedicated experimental effort needed

162 challenges for experiment study the simplest system identify quantities that can be measured with precision of a few percent and have clear theoretical interpretation precise thermometer that does not destroy probe same for density