Functional Functional renormalization –renormalization –

concepts and prospectsconcepts and prospects

physics at different physics at different length scaleslength scales

microscopic theories : where the laws are microscopic theories : where the laws are formulatedformulated

effective theories : where observations effective theories : where observations are madeare made

effective theory may involve different effective theory may involve different degrees of freedom as compared to degrees of freedom as compared to microscopic theorymicroscopic theory

example: the motion of the earth around example: the motion of the earth around the sun does not need an understanding the sun does not need an understanding of nuclear burning in the sunof nuclear burning in the sun

QCD :QCD :Short and long distance Short and long distance degrees of freedom are degrees of freedom are different !different !

Short distances : quarks and gluonsShort distances : quarks and gluons Long distances : baryons and mesonsLong distances : baryons and mesons

How to make the transition?How to make the transition?

confinement/chiral symmetry confinement/chiral symmetry breakingbreaking

collective collective degrees of freedomdegrees of freedom

Hubbard modelHubbard model

Electrons on a cubic latticeElectrons on a cubic lattice

here : on planes ( d = 2 )here : on planes ( d = 2 )

Repulsive local interaction if two Repulsive local interaction if two electrons are on the same siteelectrons are on the same site

Hopping interaction between two Hopping interaction between two neighboring sitesneighboring sites

In solid state physics : In solid state physics : “ model for everything ““ model for everything “

AntiferromagnetismAntiferromagnetism High THigh Tcc superconductivity superconductivity Metal-insulator transitionMetal-insulator transition FerromagnetismFerromagnetism

Antiferromagnetism Antiferromagnetism in d=2 Hubbard modelin d=2 Hubbard model

temperature in units of t

antiferro-magnetic orderparameter Tc/t = 0.115

U/t = 3

T.Baier,E.Bick,…

Collective degrees of Collective degrees of freedom freedom

are crucial !are crucial !for T < Tfor T < Tc c

nonvanishing order parameternonvanishing order parameter

gap for fermionsgap for fermions

low energy excitations:low energy excitations: antiferromagnetic spin wavesantiferromagnetic spin waves

effective theory / effective theory / microscopic theorymicroscopic theory

sometimes only distinguished by sometimes only distinguished by different values of couplingsdifferent values of couplings

sometimes different degrees of sometimes different degrees of freedomfreedom

Functional Renormalization Functional Renormalization GroupGroup

describes flow of effective action from describes flow of effective action from small to large length scalessmall to large length scales

perturbative renormalization : case perturbative renormalization : case where only couplings change , and where only couplings change , and couplings are smallcouplings are small

How to come from quarks and How to come from quarks and gluons to baryons and mesons ?gluons to baryons and mesons ?How to come from electrons to How to come from electrons to

spin waves ?spin waves ?

Find effective description where relevant Find effective description where relevant degrees of freedom depend on degrees of freedom depend on momentum scale or resolution in spacemomentum scale or resolution in space..

Microscope with variable resolution:Microscope with variable resolution: High resolution , small piece of volume:High resolution , small piece of volume: quarks and gluonsquarks and gluons Low resolution, large volume : hadronsLow resolution, large volume : hadrons

/

Wegner, Houghton

effective average actioneffective average action

Effective average potential :Effective average potential :Unified picture for scalar field Unified picture for scalar field

theoriestheorieswith symmetry O(N)with symmetry O(N)

in arbitrary dimension d and in arbitrary dimension d and arbitrary Narbitrary N

linear or nonlinear sigma-model forlinear or nonlinear sigma-model for

chiral symmetry breaking in QCDchiral symmetry breaking in QCD

or:or:

scalar model for antiferromagnetic spin wavesscalar model for antiferromagnetic spin waves

(linear O(3) – model )(linear O(3) – model )

fermions will be added fermions will be added laterlater

Effective potential includes Effective potential includes allall fluctuations fluctuations

Scalar field theoryScalar field theory

Flow equation for average Flow equation for average potentialpotential

Simple one loop structure –Simple one loop structure –nevertheless (almost) exactnevertheless (almost) exact

Infrared cutoffInfrared cutoff

Partial Partial differential differential equation for equation for

function U(k,function U(k,φφ) ) depending on depending on

two ( or more ) two ( or more ) variablesvariables

Z Z kk = c k-η

RegularisationRegularisation

For suitable RFor suitable Rkk ::

Momentum integral is ultraviolet and Momentum integral is ultraviolet and infrared finiteinfrared finite

Numerical integration possibleNumerical integration possible Flow equation defines a Flow equation defines a

regularization scheme ( ERGE –regularization scheme ( ERGE –regularization )regularization )

Integration by momentum shellsIntegration by momentum shells

Momentum integralMomentum integral

is dominated by is dominated by

qq22 ~ ~ k k2 2 ..

Flow only sensitive Flow only sensitive toto

physics at scale kphysics at scale k

Wave function renormalization Wave function renormalization and anomalous dimensionand anomalous dimension

for Zfor Zk k ((φφ,q,q22) : flow equation is) : flow equation is exact !exact !

Flow of effective potentialFlow of effective potential

Ising modelIsing model CO2

TT** =304.15 K =304.15 K

pp** =73.8.bar =73.8.bar

ρρ** = 0.442 g cm-2 = 0.442 g cm-2

Experiment :

S.Seide …

Critical exponents

Critical exponents , d=3Critical exponents , d=3

Solution of partial differential Solution of partial differential equation :equation :

yields highly nontrivial non-perturbative results despite the one loop structure !

Example: Kosterlitz-Thouless phase transition

Essential scaling : d=2,N=2Essential scaling : d=2,N=2

Flow equation Flow equation contains contains correctly the correctly the non-non-perturbative perturbative information !information !

(essential (essential scaling usually scaling usually described by described by vortices)vortices)

Von Gersdorff …

Kosterlitz-Thouless phase Kosterlitz-Thouless phase transition (d=2,N=2)transition (d=2,N=2)

Correct description of phase Correct description of phase withwith

Goldstone boson Goldstone boson

( infinite correlation ( infinite correlation length ) length )

for T<Tfor T<Tcc

Running renormalized d-wave Running renormalized d-wave superconducting order parameter superconducting order parameter κκ in in

Hubbard modelHubbard model

κ

- ln (k/Λ)

Tc

T>Tc

T<Tc

C.Krahl,… macroscopic scale 1 cm

Renormalized order parameter Renormalized order parameter κκ and gap in electron and gap in electron

propagator propagator ΔΔ

100 Δ / t

κ

T/Tc

jump

Temperature dependent anomalous Temperature dependent anomalous dimension dimension ηη

T/Tc

η

Effective average actionEffective average action

and and

exact renormalization group exact renormalization group equationequation

Generating functionalGenerating functional

Loop expansion :perturbation theory withinfrared cutoffin propagator

Effective average actionEffective average action

Quantum effective actionQuantum effective action

Exact renormalization Exact renormalization group equationgroup equation

Proof of Proof of exact flow equationexact flow equation

TruncationsTruncations

Functional differential equation –Functional differential equation –

cannot be solved exactlycannot be solved exactly

Approximative solution by Approximative solution by truncationtruncation ofof

most general form of effective most general form of effective actionaction

Exact flow equation for Exact flow equation for effective potentialeffective potential

Evaluate exact flow equation for Evaluate exact flow equation for homogeneous field homogeneous field φφ . .

R.h.s. involves exact propagator in R.h.s. involves exact propagator in homogeneous background field homogeneous background field φφ..

Flow equation for average Flow equation for average potentialpotential

QCD :QCD :Short and long distance Short and long distance degrees of freedom are degrees of freedom are different !different !

Short distances : quarks and gluonsShort distances : quarks and gluons Long distances : baryons and mesonsLong distances : baryons and mesons

How to make the transition?How to make the transition?

confinement/chiral symmetry confinement/chiral symmetry breakingbreaking

Nambu Jona-Lasinio modelNambu Jona-Lasinio model

……and more general quark meson modelsand more general quark meson models

Chiral condensate (NChiral condensate (Nff=2)=2)

Berges,Jungnickel,…

Critical temperature , NCritical temperature , Nf f = 2= 2

J.Berges,D.Jungnickel,…

Lattice simulation

temperature temperature dependent dependent

massesmasses pion masspion mass

sigma masssigma mass

CriticalCriticalequationequation

ofofstatestate

ScalingScalingformform

ofofequationequationof stateof state

Berges,Tetradis,…

Universal critical equation of stateis valid near critical temperature if the only light degrees of freedomare pions + sigma withO(4) – symmetry.

Not necessarily valid in QCD, even for two flavors !

conclusionsconclusions

Flow equation for effective average Flow equation for effective average action: action:

Does it work?Does it work? Why does it work?Why does it work? When does it work?When does it work? How accurately does it work?How accurately does it work?

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