Post on 10-Feb-2016
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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