# Unification from Functional Renormalization Unification from Functional Renormalization

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Unification from Functional Renormalization

Unification from Functional Renormalization n n n n 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

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

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

unified description of scalar models for all d and N

Scalar field theory

Flow equation for average potential

Simple one loop structure – nevertheless (almost) exact

Infrared cutoff

Wave function renormalization and anomalous dimension for Zk (φ, q 2) : 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 n choose N n choose d n choose initial form of potential n run ! ( quantitative results : systematic derivative expansion in second order in derivatives )

Flow of effective potential Ising model CO 2 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

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

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 n n 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 (-type ) model T<Tc κ location of minimum of u Tc local disorder pseudo gap 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

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

Exact renormalization group equation 15 years getting adult. . .

some history … ( the parents ) n n exact RG equations : Symanzik eq. , Wilson eq. , Wegner-Houghton eq. , Polchinski eq. , mathematical physics 1 PI : RG for 1 PI-four-point function and hierarchy Weinberg formal Legendre transform of Wilson eq. Nicoll, Chang n non-perturbative flow : d=3 : sharp cutoff , no wave function renormalization or momentum dependence Hasenfratz 2

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

qualitative changes that make nonperturbative 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

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 !

qualitative changes that make nonperturbative physics accessible : ( 3 ) only physics in small momentum range around k matters for the flow ERGE regularization simplementation on lattice artificial non-analyticities can be avoided

qualitative changes that make nonperturbative 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”

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

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

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

including fermions : no particular problem !

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

changing degrees of freedom

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

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 n insert identity in functional integral ( Hubbard-Stratonovich transformation ) n replace four fermion interaction by equivalent bosonic interaction ( e. g. mass and Yukawa terms) n problem : decomposition of fermion interaction into bilinears not unique ( Grassmann variables) n

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

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

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.

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, …

Bosonisation and the mean field ambiguity

Bosonic fluctuations fermion loops mean field theory boson loops

Bosonisation n adapt bosonisation 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 Bosonisation Choose αk in order to absorb the four fermion coupling in corresponding channel

Bosonisation 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 = a 2

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 local disorder pseudo gap SSB -ln(k/t) Tc=0. 115

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 < 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 !

Mermin-Wagner theorem ? No spontaneous symmetry breaking of continuous symmetry in d=2 ! not valid in practice !

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

non – relativistic bosons S. Floerchinger , … see also N. Dupuis arbitrary d , here d=2

flow of kinetic and gradient coefficients

density depletion Bogoliubov T=0

sound velocity T=0 Bogoliubov

critical temperature depends on size of probe Tc vanishes logarithmically for infinite volume

condensate and superfluid density condensate density depends on probe size

thermodynamics for large finite systems

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

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

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

wide applications condensed matter n unified description for classical bosons CW , Tetradis , Aoki , Morikawa , Souma, Sumi , Terao , Morris , Graeter , v. Gersdorff , Litim , Berges , Mouhanna , Delamotte , Canet , Bervilliers , Blaizot , Benitez , Chatie , Mendes-Galain , Wschebor n Hubbard model Baier , Bick, …, Metzner et al, Salmhofer et al, Honerkamp et al, Krahl , Kopietz et al, Katanin , Pepin , Tsai , Strack , Husemann , Lauscher

wide applications condensed matter n quantum criticality Floerchinger , Dupuis , Sengupta , Jakubczyk , n sine- Gordon model Nagy , Polonyi n disordered systems Tissier , Tarjus , Delamotte , Canet

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

wide applications condensed matter n crossover phenomena Bornholdt , Tetradis , … n superconductivity ( scalar QED 3 ) Bergerhoff , Lola , Litim , Freire, … n non equilibrium systems Delamotte , Tissier , Canet , Pietroni , Meden , Schoeller , Gasenzer , Pawlowski , Berges , Pletyukov , Reininghaus

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

wide applications ultracold atoms n Feshbach resonances Diehl, Krippa, Birse , Gies, Pawlowski , Floerchinger , Scherer , Krahl , n BEC Blaizot, Wschebor, Dupuis, Sengupta, Floerchinger

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