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Functional renormalization group for the effective average action wide applications particle physics 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, wide applications gravity asymptotic safety Reuter, Lauscher, Schwindt et al, Percacci et al, Litim, Fischer 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,…, Metzner et al, Salmhofer et al, Honerkamp et al, Krahl, disordered systems Tissier, Tarjus ,Delamotte, Canet wide applications condensed matter equation of state for CO2 liquid He4 frustrated magnets nucleation and first order phase transitions Gollisch,… Seide,… and He3 Kindermann,… Delamotte, Mouhanna, Tissier Tetradis, Strumia,…, Berges,… wide applications condensed matter crossover phenomena superconductivity ( scalar QED3 ) Bornholdt, Tetradis,… Bergerhoff, Lola, Litim , Freire,… non equilibrium systems Delamotte, Tissier, Canet, Pietroni wide applications nuclear physics effective NJL- type models Ellwanger, Jungnickel, Berges, Tetradis,…, Pirner, Schaefer, Wambach, Kunihiro, Schwenk, di-neutron condensates Birse, Krippa, equation of state for nuclear matter Berges, Jungnickel …, Birse, Krippa wide applications ultracold atoms Feshbach resonances Diehl, Gies, Pawlowski ,…, Krippa, BEC Blaizot, Wschebor, Dupuis, Sengupta unified description of scalar models for all d and N Flow equation for average potential Scalar field theory Simple one loop structure – nevertheless (almost) exact Infrared cutoff Wave function renormalization and anomalous dimension for Zk (φ,q2) : flow equation is exact ! 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 … unified approach choose N choose d choose initial form of potential run ! Flow of effective potential Ising model CO2 Experiment : S.Seide … T* =304.15 K p* =73.8.bar ρ* = 0.442 g cm-2 Critical exponents Critical exponents , d=3 ERGE world ERGE world Solution of partial differential equation : yields highly nontrivial non-perturbative results despite the one loop structure ! Example: Kosterlitz-Thouless phase transition Essential scaling : d=2,N=2 Flow equation contains correctly the nonperturbative information ! (essential scaling usually described by vortices) Von Gersdorff … Kosterlitz-Thouless phase transition (d=2,N=2) Correct description of phase with Goldstone boson ( infinite correlation length ) for T<Tc Running renormalized d-wave superconducting order parameter κ in doped Hubbard model T<Tc κ Tc T>Tc C.Krahl,… - ln (k/Λ) macroscopic scale 1 cm Renormalized order parameter κ and gap in electron propagator Δ in doped Hubbard model 100 Δ / t κ jump T/Tc Temperature dependent anomalous dimension η η T/Tc 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 including fermions : no particular problem ! changing degrees of freedom Antiferromagnetic order in the Hubbard model A functional renormalization group study T.Baier, E.Bick, … Hubbard model Functional integral formulation next neighbor interaction U>0: repulsive local interaction External parameters T : temperature μ : chemical potential (doping ) Fermion bilinears Introduce sources for bilinears Functional variation with respect to sources J yields expectation values and correlation functions 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) Partially bosonised functional integral Bosonic integration is Gaussian or: equivalent to fermionic functional integral if solve bosonic field equation as functional of fermion fields and reinsert into action fermion – boson action fermion kinetic term boson quadratic term (“classical propagator” ) Yukawa coupling source term is now linear in the bosonic fields Mean Field Theory (MFT) Evaluate Gaussian fermionic integral in background of bosonic field , e.g. Effective potential in mean field theory Mean field phase diagram for two different choices of couplings – same U ! Tc Tc μ μ Mean field ambiguity Tc Artefact of approximation … Um= Uρ= U/2 cured by inclusion of bosonic fluctuations U m= U/3 ,Uρ = 0 μ mean field phase diagram J.Jaeckel,… Rebosonization and the mean field ambiguity Bosonic fluctuations fermion loops mean field theory boson loops Rebosonization adapt bosonization to every scale k such that k-dependent field redefinition is translated to bosonic interaction H.Gies , … absorbs four-fermion coupling Modification of evolution of couplings … Evolution with k-dependent field variables Rebosonisation Choose αk such that no four fermion coupling is generated …cures mean field ambiguity Tc MFT HF/SD Flow eq. Uρ/t Flow equation for the Hubbard model T.Baier , E.Bick , …,C.Krahl Truncation Concentrate on antiferromagnetism Potential U depends only on α = a2 scale evolution of effective potential for antiferromagnetic order parameter boson contribution fermion contribution effective masses depend on α ! gap for fermions ~α running couplings Running mass term unrenormalized mass term -ln(k/t) four-fermion interaction ~ m-2 diverges dimensionless quantities renormalized antiferromagnetic order parameter κ evolution of potential minimum κ 10 -2 λ -ln(k/t) U/t = 3 , T/t = 0.15 Critical temperature For T<Tc : κ remains positive for k/t > 10-9 size of probe > 1 cm κ T/t=0.05 T/t=0.1 Tc=0.115 -ln(k/t) 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 Tc/t = 0.115 temperature in units of t Pseudo-critical temperature Tpc Limiting temperature at which bosonic mass term vanishes ( κ becomes nonvanishing ) It corresponds to a diverging four-fermion coupling This is the “critical temperature” computed in MFT ! Pseudo-gap behavior below this temperature Pseudocritical temperature Tpc MFT(HF) Flow eq. Tc μ Below the pseudocritical temperature the reign of the goldstone bosons effective nonlinear O(3) – σ - model critical behavior for interval Tc < T < Tpc evolution as for classical Heisenberg model cf. Chakravarty,Halperin,Nelson critical correlation length c,β : slowly varying functions exponential growth of correlation length compatible with observation ! at Tc : correlation length reaches sample size ! critical behavior for order parameter and correlation function Mermin-Wagner theorem ? No spontaneous symmetry breaking of continuous symmetry in d=2 ! end