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
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
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 )
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 , 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 !
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 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 )
Floerchinger, Scherer , Diehl,…see also Diehl, Gies, Pawlowski,…
BCS BEC
free bosons
interacting bosons
BCS
Gorkov
BCS – BEC crossoverBCS – BEC crossover
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 )
Floerchinger, Scherer , Diehl,…see also Diehl, Gies, Pawlowski,…
BCS BEC
free bosons
interacting bosons
BCS
Gorkov
BCS – BEC crossoverBCS – BEC crossover
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 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,
wide applicationswide applications
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
wide applicationswide applications
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
wide applicationswide applications
condensed 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 applications
condensed 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 applications
condensed 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 applications
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
wide applicationswide applications
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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