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Functional Functional renormalization renormalization group equation group equation
for strongly for strongly correlated correlated fermionsfermions
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collective collective degrees of degrees of freedomfreedom
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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
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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
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Hubbard modelHubbard model
Functional integral formulation
U > 0 : repulsive local interaction
next neighbor interaction
External parametersT : temperatureμ : chemical potential (doping )
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Fermi surfaceFermi surface
Fermion quadratic term
ωF = (2n+1)πT
Fermi surface : zeros of P for T=0
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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
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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
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QCD :QCD :
Short and long distance Short and long distance
degrees of freedom are different !degrees of freedom are 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 breakingconfinement/chiral symmetry breaking
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Nambu Jona-Lasinio modelNambu Jona-Lasinio model
……and more general quark meson modelsand more general quark meson models
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Chiral condensate Chiral condensate (N(Nff=2)=2)
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Functional Renormalization Functional Renormalization GroupGroup
from small to large scalesfrom small to large scales
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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
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Effective potential includes Effective potential includes allall fluctuations fluctuations
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Scalar field theoryScalar field theory
linear sigma-model forlinear sigma-model forchiral symmetry breaking in QCDchiral symmetry breaking in QCDor:or:scalar model for antiferromagnetic spin scalar model for antiferromagnetic spin
waveswaves(linear O(3) – model )(linear O(3) – model )
fermions will be added fermions will be added laterlater
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Scalar field theoryScalar field theory
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Flow equation for average Flow equation for average potentialpotential
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Simple one loop structure –Simple one loop structure –nevertheless (almost) exactnevertheless (almost) exact
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Infrared cutoffInfrared cutoff
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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
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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 )
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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
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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 !
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Effective average actionEffective average action
and and
exact renormalization group exact renormalization group equationequation
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Generating functionalGenerating functional
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Loop expansion :perturbation theory withinfrared cutoffin propagator
Effective average actionEffective average action
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Quantum effective actionQuantum effective action
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Exact renormalization Exact renormalization group equationgroup equation
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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..
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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
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AntiferromagneticAntiferromagnetic order in the order in the
Hubbard modelHubbard model
A functional renormalization A functional renormalization group studygroup study
T.Baier, E.Bick, …
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Temperature dependence of Temperature dependence of antiferromagnetic order antiferromagnetic order
parameterparameter
temperature in units of t
antiferro-magnetic orderparameter Tc/t = 0.115
U = 3
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Mermin-Wagner theorem Mermin-Wagner theorem ??
NoNo spontaneous symmetry breaking spontaneous symmetry breaking
of continuous symmetry in of continuous symmetry in d=2 d=2 !!
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Fermion bilinearsFermion bilinears
Introduce sources for bilinears
Functional variation withrespect to sources Jyields expectation valuesand correlation functions
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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)
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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
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fermion – boson actionfermion – boson action
fermion kinetic term
boson quadratic term (“classical propagator” )
Yukawa coupling
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source termsource term
is now linear in the bosonic fields
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Mean Field Theory (MFT)Mean Field Theory (MFT)
Evaluate Gaussian fermionic integralin background of bosonic field , e.g.
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Effective potential in mean Effective potential in mean field theoryfield theory
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Mean field phase Mean field phase diagramdiagram
μμ
TcTc
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Mean field inverse Mean field inverse propagatorpropagator
for spin wavesfor spin waves
T/t = 0.5 T/t = 0.15
Pm(q) Pm(q)
Baier,Bick,…
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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,…
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Flow equationFlow equationfor thefor the
Hubbard modelHubbard model
T.Baier , E.Bick , …
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TruncationTruncationConcentrate on antiferromagnetism
Potential U depends only on α = a2
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scale evolution of effective scale evolution of effective potential for potential for
antiferromagnetic order antiferromagnetic order parameterparameter
boson contribution
fermion contribution
effective masses depend on α !
gap for fermions ~α
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running couplingsrunning couplings
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unrenormalized mass term
Running mass termRunning mass term
four-fermion interaction ~ m-2 diverges
-ln(k/t)
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dimensionless quantitiesdimensionless quantities
renormalized antiferromagnetic order parameter κ
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evolution of potential evolution of potential minimumminimum
-ln(k/t)
U/t = 3 , T/t = 0.15
κ
10 -2 λ
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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
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Pseudocritical Pseudocritical temperature Ttemperature Tpcpc
Limiting temperature at which bosonic Limiting temperature at which bosonic mass term vanishes ( mass term vanishes ( κκ becomes becomes nonvanishing ) nonvanishing )
It corresponds to a diverging four-It corresponds to a diverging four-fermion couplingfermion coupling
This is the “critical temperature” This is the “critical temperature” computed in MFT !computed in MFT !
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Pseudocritical Pseudocritical temperaturetemperature
Tpc
μ
Tc
MFT(HF)
Flow eq.
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critical behaviorcritical behavior
for interval Tc < T < Tpc
evolution as for classical Heisenberg model
cf. Chakravarty,Halperin,Nelson
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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 !
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critical behavior for order critical behavior for order parameter and correlation parameter and correlation
functionfunction
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Bosonic fluctuationsBosonic fluctuations
fermion loops boson loops
mean field theory
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RebosonisationRebosonisation
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
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Modification of evolution of Modification of evolution of couplings …couplings …
Choose αk such that nofour fermion coupling is generated
Evolution with k-dependentfield variables
Rebosonisation
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……cures mean field cures mean field ambiguityambiguity
Tc
Uρ/t
MFT
Flow eq.
HF/SD
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Nambu Jona-Lasinio modelNambu Jona-Lasinio model
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Critical temperature , NCritical temperature , Nf f = 2= 2
J.Berges,D.Jungnickel,…
Lattice simulation
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Chiral condensateChiral condensate
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temperature temperature dependent dependent
massesmasses pion masspion mass
sigma masssigma mass
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CriticalCriticalequationequation
ofofstatestate
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ScalingScalingformform
ofofequationequationof stateof state
Berges,Tetradis,…
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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 !
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