Emergence of new laws Emergence of new laws withwith

Functional Functional Renormalization Renormalization

different laws at different different laws at different scalesscales

fluctuations wash out many details fluctuations wash out many details of microscopic lawsof microscopic laws

new structures as bound states or new structures as bound states or collective phenomena emergecollective phenomena emerge

elementary particles – earth – elementary particles – earth – Universe :Universe :

key problem in Physics !key problem in Physics !

scale dependent lawsscale dependent laws

scale dependent ( running or flowing scale dependent ( running or flowing ) couplings) couplings

flowing functionsflowing functions

flowing functionalsflowing functionals

flowing actionflowing action

Wikipedia

flowing actionflowing action

microscopic law

macroscopic law

infinitely many couplings

effective theorieseffective theories

planetsplanets fundamental microscopic law for fundamental microscopic law for

matter in solar system:matter in solar system: Schroedinger equation for many electrons Schroedinger equation for many electrons

and nucleons,and nucleons, in gravitational potential of sunin gravitational potential of sun with electromagnetic and gravitational with electromagnetic and gravitational

interactions (strong and weak interactions interactions (strong and weak interactions neglected) neglected)

effective theory for effective theory for planetsplanets

at long distances , large time scales :at long distances , large time scales : point-like planets , only mass of planets point-like planets , only mass of planets

plays a roleplays a role effective theory : Newtonian mechanics effective theory : Newtonian mechanics

for point particlesfor point particles loss of memoryloss of memory new simple lawsnew simple laws only a few parameters : masses of planetsonly a few parameters : masses of planets determined by microscopic parameters + determined by microscopic parameters +

historyhistory

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

functional functional renormalizationrenormalization

transition from microscopic to transition from microscopic to effective theory is made continuouseffective theory is made continuous

effective laws depend on scale keffective laws depend on scale k flow in space of theoriesflow in space of theories flow from simplicity to complexity – flow from simplicity to complexity –

if theory is simple for large kif theory is simple for large k or opposite , if theory gets simple for or opposite , if theory gets simple for

small ksmall k

Scales in strong interactionsScales in strong interactions

simple

complicated

simple

flow of functionsflow of functions

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

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 !

Scaling form of evolution Scaling form of evolution equationequation

Tetradis …

On r.h.s. :neither the scale k nor the wave function renormalization Z appear explicitly.

Scaling solution:no dependence on t;correspondsto second order phase transition.

unified approachunified approach

choose Nchoose N choose dchoose d choose initial form of potentialchoose initial form of potential run !run !

( quantitative results : systematic derivative ( quantitative results : systematic derivative expansion in second order in derivatives )expansion in second order in derivatives )

unified description of unified description of scalar models for all d scalar models for all d

and Nand N

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 , BMW critical exponents , BMW approximationapproximation

Blaizot, Benitez , … , Wschebor

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

Temperature dependent anomalous Temperature dependent anomalous dimension dimension ηη

T/Tc

η

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

doped Hubbard (-type ) modeldoped Hubbard (-type ) model

κ

- ln (k/Λ)

Tc

T>Tc

T<Tc

C.Krahl,… macroscopic scale 1 cm

locationofminimumof u

local disorderpseudo gap

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

propagator propagator ΔΔin doped Hubbard modelin doped Hubbard model

100 Δ / t

κ

T/Tc

jump

unificationunification

complexity

abstract laws

quantum gravity

grand unification

standard model

electro-magnetism

gravity

Landau universal functionaltheory critical physics renormalization

flow of functionalsflow of functionals

f(x) f [φ(x)]

Exact renormalization Exact renormalization group equationgroup equation

some history … ( the some history … ( the parents )parents )

exact RG equations :exact RG equations : Symanzik eq. , Wilson eq. , Wegner-Houghton eq. , Symanzik eq. , Wilson eq. , Wegner-Houghton eq. ,

Polchinski eq. ,Polchinski eq. , mathematical physicsmathematical physics

1PI :1PI : RG for 1PI-four-point function and hierarchy RG for 1PI-four-point function and hierarchy WeinbergWeinberg

formal Legendre transform of Wilson eq.formal Legendre transform of Wilson eq. Nicoll, ChangNicoll, Chang

non-perturbative flow :non-perturbative flow : d=3 : sharp cutoff , d=3 : sharp cutoff , no wave function renormalization or no wave function renormalization or

momentum dependencemomentum dependence HasenfratzHasenfratz22

flow equations and composite degrees of

freedom

Flowing quark Flowing quark interactionsinteractions

U. Ellwanger,… Nucl.Phys.B423(1994)137

Flowing four-quark Flowing four-quark vertexvertex

emergence of mesons

Floerchinger, Scherer , Diehl,…see also Diehl, Floerchinger, Gies, Pawlowski,…

BCS BEC

free bosons

interacting bosons

BCS

Gorkov

BCS – BEC crossoverBCS – BEC crossover

changing degrees of changing degrees of freedomfreedom

Anti-ferromagnetic Anti-ferromagnetic order in the Hubbard order in the Hubbard

modelmodel

transition from transition from

microscopic theory for fermions to microscopic theory for fermions to macroscopic theory for bosonsmacroscopic theory for bosons

T.Baier, E.Bick, …C.Krahl, J.Mueller, S.Friederich

Hubbard modelHubbard model

Functional integral formulation

U > 0 : repulsive local interaction

next neighbor interaction

External parametersT : temperatureμ : chemical potential (doping )

Fermion bilinearsFermion bilinears

Introduce sources for bilinears

Functional variation withrespect to sources Jyields expectation valuesand correlation functions

Partial BosonisationPartial Bosonisation collective bosonic variables for fermion collective bosonic variables for fermion

bilinearsbilinears insert identity in functional integralinsert identity in functional integral ( Hubbard-Stratonovich transformation )( Hubbard-Stratonovich transformation ) replace four fermion interaction by replace four fermion interaction by

equivalent bosonic interaction ( e.g. mass equivalent bosonic interaction ( e.g. mass and Yukawa terms)and Yukawa terms)

problem : decomposition of fermion problem : decomposition of fermion interaction into bilinears not unique interaction into bilinears not unique ( Grassmann variables)( Grassmann variables)

Partially bosonised functional Partially bosonised functional integralintegral

equivalent to fermionic functional integral

if

Bosonic integrationis Gaussian

or:

solve bosonic field equation as functional of fermion fields and reinsert into action

more bosons …more bosons …

additional fields may be added formally :additional fields may be added formally :

only mass term + source term : only mass term + source term : decoupled bosondecoupled boson

introduction of boson fields not linked to introduction of boson fields not linked to Hubbard-Stratonovich transformationHubbard-Stratonovich transformation

fermion – boson actionfermion – boson action

fermion kinetic term

boson quadratic term (“classical propagator” )

Yukawa coupling

source termsource term

is now linear in the bosonic fields

effective action treats fermions and composite bosons on equal footing !

Mean Field Theory (MFT)Mean Field Theory (MFT)

Evaluate Gaussian fermionic integralin background of bosonic field , e.g.

Mean field phase Mean field phase diagramdiagram

μμ

TcTc

for two different choices of couplings – same U !

Mean field ambiguityMean field ambiguity

Tc

μ

mean field phase diagram

Um= Uρ= U/2

U m= U/3 ,Uρ = 0

Artefact of approximation …

cured by inclusion ofbosonic fluctuations

J.Jaeckel,…

partial bosonisation and the partial bosonisation and the

mean field ambiguitymean field ambiguity

Bosonic fluctuationsBosonic fluctuations

fermion loops boson loops

mean field theory

flowing bosonisationflowing bosonisation

adapt bosonisation adapt bosonisation to every scale k to every scale k such thatsuch that

is translated to is translated to bosonic interactionbosonic interaction

H.Gies , …

k-dependent field redefinition

absorbs four-fermion coupling

flowing bosonisationflowing bosonisation

Choose αk in order to absorb the four fermion coupling in corresponding channel

Evolution with k-dependentfield variables

modified flow of couplings

Bosonisation Bosonisation cures mean field ambiguitycures mean field ambiguity

Tc

Uρ/t

MFT

Flow eq.

HF/SD

Flow equationFlow equationfor thefor the

Hubbard modelHubbard model

T.Baier , E.Bick , …,C.Krahl, J.Mueller, S.Friederich

Below the critical Below the critical temperature :temperature :

temperature in units of t

antiferro-magnetic orderparameter

Tc/t = 0.115

U = 3

Infinite-volume-correlation-length becomes larger than sample size

finite sample ≈ finite k : order remains effectively

Critical temperatureCritical temperatureFor T<Tc : κ remains positive for k/t > 10-9

size of probe > 1 cm

-ln(k/t)

κ

Tc=0.115

T/t=0.05

T/t=0.1

local disorderpseudo gap

SSB

Mermin-Wagner theorem Mermin-Wagner theorem ??

NoNo spontaneous symmetry spontaneous symmetry breaking breaking

of continuous symmetry in of continuous symmetry in d=2 d=2 !!

not valid in practice !not valid in practice !

Pseudo-critical Pseudo-critical temperature Ttemperature Tpcpc

Limiting temperature at which bosonic mass Limiting temperature at which bosonic mass term vanishes ( term vanishes ( κκ becomes nonvanishing ) becomes nonvanishing )

It corresponds to a diverging four-fermion It corresponds to a diverging four-fermion couplingcoupling

This is the “critical temperature” computed This is the “critical temperature” computed in MFT !in MFT !

Pseudo-gap behavior below this temperaturePseudo-gap behavior below this temperature

Pseudocritical Pseudocritical temperaturetemperature

Tpc

μ

Tc

MFT(HF)

Flow eq.

Below the pseudocritical Below the pseudocritical temperaturetemperature

the reign of the goldstone bosons

effective nonlinear O(3) – σ - model

Critical temperatureCritical temperature

-ln(k/t)

κ

Tc=0.115

T/t=0.05

T/t=0.1

local disorderpseudo gap

SSB

only Goldstonebosons matter !

critical behaviorcritical behavior

for interval Tc < T < Tpc

evolution as for classical Heisenberg model cf. Chakravarty,Halperin,Nelson

dimensionless coupling of non-linear sigma-model : g2 ~ κ -1

two-loop beta function for g

effective theory

non-linear O(3)-sigma-model asymptotic freedom

from fermionic microscopic law to bosonic macroscopic law

transition to linear sigma-model

large coupling regime of non-linear sigma-model :

small renormalized order parameter κ

transition to symmetric phase

again change of effective laws : linear sigma-model is simple , strongly coupled non-linear sigma-model is

complicated

critical correlation critical correlation lengthlength

c,β : slowly varying functions

exponential growth of correlation length compatible with observation !

at Tc : correlation length reaches sample size !

conclusion

functional renormalization offers an efficient method for adding new relevant degrees of freedom or removing irrelevant degrees of freedom

continuous description of the emergence of new laws

Unification fromUnification fromFunctional Renormalization Functional Renormalization

fluctuations in d=0,1,2,3,...fluctuations in d=0,1,2,3,... linear and non-linear sigma modelslinear and non-linear sigma models vortices and perturbation theoryvortices and perturbation theory bosonic and fermionic modelsbosonic and fermionic models relativistic and non-relativistic physicsrelativistic and non-relativistic physics classical and quantum statisticsclassical and quantum statistics non-universal and universal aspectsnon-universal and universal aspects homogenous systems and local disorderhomogenous systems and local disorder equilibrium and out of equilibriumequilibrium and out of equilibrium

end

unification:unification:functional integral / flow functional integral / flow

equationequation

simplicity of average actionsimplicity of average action explicit presence of scaleexplicit presence of scale differentiating is easier than differentiating is easier than

integrating…integrating…

qualitative changes that make qualitative changes that make non-perturbative physics non-perturbative physics

accessible :accessible :

( 1 ) basic object is ( 1 ) basic object is simplesimple

average action ~ classical actionaverage action ~ classical action

~ generalized Landau ~ generalized Landau theorytheory

direct connection to thermodynamicsdirect connection to thermodynamics

(coarse grained free energy )(coarse grained free energy )

qualitative changes that make qualitative changes that make non-perturbative physics non-perturbative physics

accessible :accessible :

( 2 ) Infrared scale k ( 2 ) Infrared scale k

instead of Ultraviolet instead of Ultraviolet cutoff cutoff ΛΛ

short distance memory not lostshort distance memory not lost

no modes are integrated out , but only part no modes are integrated out , but only part of the fluctuations is includedof the fluctuations is included

simple one-loop form of flowsimple one-loop form of flow

simple comparison with perturbation theorysimple comparison with perturbation theory

infrared cutoff kinfrared cutoff k

cutoff on momentum cutoff on momentum resolution resolution

or frequency or frequency resolutionresolution

e.g. distance from pure anti-ferromagnetic e.g. distance from pure anti-ferromagnetic momentum or from Fermi surfacemomentum or from Fermi surface

intuitive interpretation of k by intuitive interpretation of k by association with physical IR-cutoff , association with physical IR-cutoff , i.e. finite size of system :i.e. finite size of system :

arbitrarily small momentum arbitrarily small momentum differences cannot be resolved !differences cannot be resolved !

qualitative changes that make qualitative changes that make non-perturbative physics non-perturbative physics

accessible :accessible :

( 3 ) only physics in small ( 3 ) only physics in small momentum range around k momentum range around k matters for the flowmatters for the flow

ERGE regularizationERGE regularization

simple implementation on latticesimple implementation on lattice

artificial non-analyticities can be avoidedartificial non-analyticities can be avoided

qualitative changes that make qualitative changes that make non-perturbative physics non-perturbative physics

accessible :accessible :

( 4 ) ( 4 ) flexibilityflexibility

change of fields change of fields

microscopic or composite variablesmicroscopic or composite variables

simple description of collective degrees of freedom simple description of collective degrees of freedom and bound statesand bound states

many possible choices of “cutoffs”many possible choices of “cutoffs”