Unification from Functional Renormalization

Unification fromUnification fromFunctional Functional

Renormalization Renormalization

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

unificationunification

complexity

abstract laws

quantum gravity

grand unification

standard model

electro-magnetism

gravity

Landau universal functionaltheory critical physics renormalization

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…

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

and Nand N

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 )

Flow of effective potentialFlow of effective potential

Ising modelIsing model CO2

TT** =304.15 K =304.15 Kpp** =73.8.bar =73.8.barρρ** = 0.442 g cm-2 = 0.442 g cm-2

Experiment :

S.Seide …

Critical exponents

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

ERGE world ERGE world

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

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

Temperature dependent anomalous Temperature dependent anomalous dimension dimension ηη

T/Tc

η

Unification fromUnification fromFunctional Renormalization Functional Renormalization ☺☺fluctuations in d=0,1,2,3,4,...fluctuations in d=0,1,2,3,4,...☺☺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

Exact renormalization Exact renormalization group equationgroup equation

15 years

getting adult...

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 momentum no wave function renormalization or momentum

dependencedependence HasenfratzHasenfratz22

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 theory~ generalized Landau theory

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 lostno modes are integrated out , but only part no modes are integrated out , but only part

of the fluctuations is includedof the fluctuations is includedsimple one-loop form of flowsimple one-loop form of flowsimple comparison with perturbation theorysimple comparison with perturbation theory

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

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

Proof of Proof of exact flow equationexact flow equation

sources j canmultiply arbitraryoperators

φ : associated fields

TruncationsTruncations

Functional differential equation –Functional differential equation – cannot be solved exactlycannot be solved exactlyApproximative solution by Approximative solution by truncationtruncation ofof most general form of effective most general form of effective

actionaction

convergence and errorsconvergence and errors apparent fast convergence : no apparent fast convergence : no

series resummationseries resummation rough error estimate by different rough error estimate by different

cutoffs and truncations , Fierz cutoffs and truncations , Fierz ambiguity etc.ambiguity etc.

in general : understanding of physics in general : understanding of physics crucial crucial

no standardized procedureno standardized procedure

including fermions :including fermions :

no particular problem !no particular problem !

Floerchinger, Scherer , Diehl,…see also Diehl, 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 order in the

Hubbard modelHubbard model

A functional renormalization A functional renormalization group studygroup study

T.Baier, E.Bick, …C.Krahl

Hubbard modelHubbard modelFunctional 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 equivalent replace four fermion interaction by equivalent

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

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,…

Bosonisation and the Bosonisation and the mean field ambiguitymean field ambiguity

Bosonic fluctuationsBosonic fluctuations

fermion loops boson loops

mean field theory

BosonisationBosonisation 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

Modification of evolution of Modification of evolution of couplings …couplings …

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

Evolution with k-dependentfield variables

Bosonisation

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

TruncationTruncationConcentrate on antiferromagnetism

Potential U depends only on α = a2

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

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

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 in This is the “critical temperature” computed in MFT !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 behaviorcritical behaviorfor interval Tc < T < Tpc

evolution as for classical Heisenberg model

cf. Chakravarty,Halperin,Nelson

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 !

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 !

Unification fromUnification fromFunctional Renormalization Functional Renormalization fluctuations in d=0,1,2,3,4,...fluctuations in d=0,1,2,3,4,...☺☺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

non – relativistic bosonsnon – relativistic bosons

S. Floerchinger , …see also N. Dupuis

arbitrary d , here d=2

flow of kinetic and gradient flow of kinetic and gradient coefficientscoefficients

density depletiondensity depletion

T=0Bogoliubov

sound velocitysound velocity

T=0

Bogoliubov

T=0

critical temperaturecritical temperaturedepends on size of probedepends on size of probe

Tc vanishes logarithmically for infinite volume

condensate and superfluid condensate and superfluid densitydensity

superfluid density

condensate densitydepends on probe size

thermodynamics thermodynamics forfor

large finite systemslarge finite systems

Unification fromUnification fromFunctional Renormalization Functional Renormalization fluctuations in d=0,1,2,3,4,...fluctuations in d=0,1,2,3,4,... 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

wide applicationswide applicationsparticle physicsparticle physics

gauge theories, QCDgauge theories, QCD Reuter,…, Marchesini et al, Ellwanger et al, Litim, Reuter,…, Marchesini et al, Ellwanger et al, Litim,

Pawlowski, Gies ,Freire, Morris et al., Braun , many othersPawlowski, Gies ,Freire, Morris et al., Braun , many others

electroweak interactions, gauge hierarchyelectroweak interactions, gauge hierarchy problemproblem

Jaeckel,Gies,…Jaeckel,Gies,… electroweak phase transitionelectroweak phase transition Reuter,Tetradis,…Bergerhoff,Reuter,Tetradis,…Bergerhoff,

wide applicationswide applicationsgravitygravity

asymptotic safetyasymptotic safety Reuter, Lauscher, Schwindt et al, Percacci et al, Reuter, Lauscher, Schwindt et al, Percacci et al,

Litim, Fischer,Litim, Fischer, SaueressigSaueressig

wide applicationswide applicationscondensed mattercondensed matter

unified description for classical bosons unified description for classical bosons CW , Tetradis , Aoki , Morikawa , Souma, Sumi , CW , Tetradis , Aoki , Morikawa , Souma, Sumi ,

Terao , Morris , Graeter , v.Gersdorff , Litim , Terao , Morris , Graeter , v.Gersdorff , Litim , Berges , Mouhanna , Delamotte , Canet , Bervilliers , Berges , Mouhanna , Delamotte , Canet , Bervilliers , Blaizot , Benitez , Chatie , Mendes-Galain , Blaizot , Benitez , Chatie , Mendes-Galain , Wschebor Wschebor

Hubbard model Hubbard model Baier , Bick,…, Metzner et al, Salmhofer et al, Baier , Bick,…, Metzner et al, Salmhofer et al,

Honerkamp et al, Krahl , Kopietz et al, Katanin , Honerkamp et al, Krahl , Kopietz et al, Katanin , Pepin , Tsai , Strack ,Pepin , Tsai , Strack ,

Husemann , Lauscher Husemann , Lauscher

wide applicationswide applicationscondensed mattercondensed matter

quantum criticalityquantum criticality Floerchinger , Dupuis , Sengupta , Jakubczyk ,Floerchinger , Dupuis , Sengupta , Jakubczyk , sine- Gordon model sine- Gordon model Nagy , PolonyiNagy , Polonyi disordered systems disordered systems Tissier , Tarjus , Delamotte , CanetTissier , Tarjus , Delamotte , Canet

wide applicationswide applicationscondensed mattercondensed matter

equation of state for COequation of state for CO2 2 Seide,…Seide,…

liquid Heliquid He44 Gollisch,…Gollisch,… and He and He3 3 Kindermann,…Kindermann,…

frustrated magnets frustrated magnets Delamotte, Mouhanna, TissierDelamotte, Mouhanna, Tissier

nucleation and first order phase transitionsnucleation and first order phase transitions Tetradis, Strumia,…, Berges,…Tetradis, Strumia,…, Berges,…

wide applicationswide applicationscondensed mattercondensed matter

crossover phenomena crossover phenomena Bornholdt , Tetradis ,…Bornholdt , Tetradis ,… superconductivity ( scalar QEDsuperconductivity ( scalar QED3 3 )) Bergerhoff , Lola , Litim , Freire,…Bergerhoff , Lola , Litim , Freire,… non equilibrium systemsnon equilibrium systems Delamotte , Tissier , Canet , Pietroni , Meden , Delamotte , Tissier , Canet , Pietroni , Meden ,

Schoeller , Gasenzer , Pawlowski , Berges , Schoeller , Gasenzer , Pawlowski , Berges , Pletyukov , Reininghaus Pletyukov , Reininghaus

wide applicationswide applicationsnuclear physicsnuclear physics

effective NJL- type modelseffective NJL- type models Ellwanger , Jungnickel , Berges , Tetradis,…, Ellwanger , Jungnickel , Berges , Tetradis,…,

Pirner , Schaefer , Wambach , Kunihiro , Schwenk Pirner , Schaefer , Wambach , Kunihiro , Schwenk di-neutron condensatesdi-neutron condensates Birse, Krippa, Birse, Krippa, equation of state for nuclear matterequation of state for nuclear matter Berges, Jungnickel …, Birse, Krippa Berges, Jungnickel …, Birse, Krippa nuclear interactionsnuclear interactions SchwenkSchwenk

wide applicationswide applicationsultracold atomsultracold atoms

Feshbach resonances Feshbach resonances Diehl, Krippa, Birse , Gies, Pawlowski , Diehl, Krippa, Birse , Gies, Pawlowski ,

Floerchinger , Scherer , Krahl , Floerchinger , Scherer , Krahl ,

BEC BEC Blaizot, Wschebor, Dupuis, Sengupta, FloerchingerBlaizot, Wschebor, Dupuis, Sengupta, Floerchinger