Unification from Functional Renormalization

Unification fromUnification fromFunctional Functional

Renormalization Renormalization

/

Wegner, Houghton

Effective potential includes Effective potential includes allall fluctuations fluctuations

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 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

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


☺☺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

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 ~ generalized Landau theorytheory

direct connection to thermodynamicsdirect connection to thermodynamics

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



( 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 , i.e. association with physical IR-cutoff , i.e. finite size of system :finite size of system :

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



( 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



( 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 exactly

Approximative 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 !

Universality in ultra-cold Universality in ultra-cold fermionic atom gasesfermionic atom gases

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

BCS BEC

free bosons

interacting bosons

BCS

Gorkov

BCS – BEC crossoverBCS – BEC crossover

BEC – BCS crossoverBEC – BCS crossover

Bound molecules of two atoms Bound molecules of two atoms on microscopic scale:on microscopic scale:

Bose-Einstein condensate (BEC ) for low TBose-Einstein condensate (BEC ) for low T

Fermions with attractive interactionsFermions with attractive interactions (molecules play no role ) :(molecules play no role ) :

BCS – superfluidity at low T BCS – superfluidity at low T by condensation of Cooper pairsby condensation of Cooper pairs

CrossoverCrossover by by Feshbach resonanceFeshbach resonance as a transition in terms of external as a transition in terms of external magnetic fieldmagnetic field

Feshbach resonanceFeshbach resonance

H.Stoof

scattering lengthscattering length

BCS

BEC

chemical potentialchemical potential

BEC

BCS

inverse scattering length

concentrationconcentration c = a kF , a(B) : scattering length needs computation of density n=kF

3/(3π2)

BCS

BEC

dilute dilutedense

T = 0

non-interactingFermi gas

non-interactingBose gas

universalityuniversality

same curve for Li and K atoms ?

BCS

BEC

dilute dilutedense

T = 0

Quantum Monte Carlo

different methodsdifferent methods

who cares about details ?who cares about details ?

a theorists game …?a theorists game …?

RGMFT

precision many body theoryprecision many body theory- quantum field theory- quantum field theory - -

so far :so far : particle physics : particle physics : perturbative calculationsperturbative calculations

magnetic moment of electron : magnetic moment of electron :

g/2 = 1.001 159 652 180 85 ( 76 ) g/2 = 1.001 159 652 180 85 ( 76 ) ( Gabrielse et ( Gabrielse et al. )al. )

statistical physics : universal critical exponents statistical physics : universal critical exponents for second order phase transitions : for second order phase transitions : νν = 0.6308 = 0.6308 (10)(10)

renormalization grouprenormalization group lattice simulationslattice simulations for bosonic systems in particle for bosonic systems in particle

and statistical physics ( e.g. QCD )and statistical physics ( e.g. QCD )


BCS BEC

free bosons

interacting bosons

BCS

Gorkov


QFT with fermionsQFT with fermions

needed:needed:

universal theoretical tools for universal theoretical tools for complex fermionic systemscomplex fermionic systems

wide applications :wide applications :

electrons in solids , electrons in solids ,

nuclear matter in neutron stars , nuclear matter in neutron stars , ….….

QFT for non-relativistic QFT for non-relativistic fermionsfermions

functional integral, actionfunctional integral, action

τ : euclidean time on torus with circumference 1/Tσ : effective chemical potential

perturbation theory:Feynman rules

variablesvariables

ψψ : Grassmann variables : Grassmann variables φφ : bosonic field with atom number : bosonic field with atom number

twotwo

What is What is φφ ? ? microscopic molecule, microscopic molecule, macroscopic Cooper pair ?macroscopic Cooper pair ?

All !All !

parametersparameters

detuning detuning νν(B)(B)

Yukawa or Feshbach coupling hYukawa or Feshbach coupling hφφ

fermionic actionfermionic action

equivalent fermionic action , in equivalent fermionic action , in general not localgeneral not local

scattering length ascattering length a

broad resonance : pointlike limitbroad resonance : pointlike limit large Feshbach couplinglarge Feshbach coupling

a= M λ/4π

parametersparameters

Yukawa or Feshbach coupling hYukawa or Feshbach coupling hφφ

scattering length ascattering length a

broad resonance :broad resonance : h hφφ drops outdrops out

concentration cconcentration c

universalityuniversality Are these parameters enough for a Are these parameters enough for a

quantitatively precise description ?quantitatively precise description ?

Have Li and K the same crossover when Have Li and K the same crossover when described with these parameters ?described with these parameters ?

Long distance physics looses memory of detailed Long distance physics looses memory of detailed microscopic properties of atoms and molecules !microscopic properties of atoms and molecules !

universality for cuniversality for c-1-1 = 0 : Ho,…( valid for broad = 0 : Ho,…( valid for broad resonance)resonance)

here: whole crossover rangehere: whole crossover range

analogy with particle analogy with particle physicsphysics

microscopic theory not known -microscopic theory not known - nevertheless “macroscopic theory” nevertheless “macroscopic theory”

characterized by a finite number of characterized by a finite number of ““renormalizable couplings”renormalizable couplings”

mmee , , αα ; g ; g ww , g , g ss , M , M w w , …, …

here : here : cc , h , hφφ ( only ( only cc for broad for broad resonance )resonance )

analogy with analogy with universal critical exponentsuniversal critical exponents

only one relevant parameter :only one relevant parameter :

T - TT - Tcc

units and dimensionsunits and dimensions

ħ =1 ; kħ =1 ; kBB = 1 = 1 momentum ~ lengthmomentum ~ length-1-1 ~ mass ~ eV ~ mass ~ eV energies : 2ME ~ (momentum)energies : 2ME ~ (momentum)22

( M : atom mass )( M : atom mass ) typical momentum unit : Fermi momentumtypical momentum unit : Fermi momentum typical energy and temperature unit : Fermi typical energy and temperature unit : Fermi

energyenergy time ~ (momentum) time ~ (momentum) -2-2

canonical dimensions different from canonical dimensions different from relativistic QFT !relativistic QFT !

rescaled actionrescaled action

M drops outM drops out all quantities in units of kall quantities in units of kF F ifif

what is to be computed ?what is to be computed ?

Inclusion of fluctuation effectsInclusion of fluctuation effectsvia via functional integralfunctional integralleads to leads to effective action.effective action.

This contains all relevant This contains all relevant information for arbitrary T information for arbitrary T and n !and n !

effective actioneffective action

integrate out all quantum and integrate out all quantum and thermal fluctuationsthermal fluctuations

quantum effective actionquantum effective action generates full propagators and generates full propagators and

verticesvertices richer structure than classical actionricher structure than classical action

effective actioneffective action

includes all quantum and thermal fluctuationsincludes all quantum and thermal fluctuations formulated here in terms of renormalized formulated here in terms of renormalized

fieldsfields involves renormalized couplingsinvolves renormalized couplings

effective potentialeffective potential

minimum determines order parameterminimum determines order parameter

condensate fractioncondensate fraction

Ωc = 2 ρ0/n

effective potentialeffective potential value of value of φφ at potential minimum : at potential minimum : order parameter , determines order parameter , determines

condensate fractioncondensate fraction second derivative of U with respect to second derivative of U with respect to

φφ yields correlation length yields correlation length derivative with respect to derivative with respect to σσ yields yields

densitydensity fourth derivative of U with respect to fourth derivative of U with respect to φφ

yields molecular scattering lengthyields molecular scattering length

renormalized fields and renormalized fields and couplingscouplings

challenge for ultra-challenge for ultra-cold atoms :cold atoms :

Non-relativistic fermion systems Non-relativistic fermion systems with precisionwith precision

similar to particle physics !similar to particle physics !

( QCD with quarks )( QCD with quarks )


BCS BEC

free bosons

interacting bosons

BCS

Gorkov



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 applications

particle 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,


gravitygravity

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

Litim, Fischer,Litim, Fischer,

SaueressigSaueressig


condensed 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



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



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,…



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


nuclear physicsnuclear physics

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

Pirner , Schaefer , Wambach , Kunihiro , Pirner , Schaefer , Wambach , Kunihiro , Schwenk 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


ultracold 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, Blaizot, Wschebor, Dupuis, Sengupta,

FloerchingerFloerchinger

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Unification from Functional Renormalization