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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”
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