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Functional Functional renormalization group renormalization group
for the effective average for the effective average actionaction
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., many othersPawlowski, Gies ,Freire, Morris et al., 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, FischerLitim, Fischer
wide applicationswide applications
condensed mattercondensed matter
unified description for classical bosons unified description for classical bosons CW, Tetradis , Aoki, Morikawa, Souma, Sumi, Terao , CW, Tetradis , Aoki, Morikawa, Souma, Sumi, Terao ,
Morris , Graeter, v.Gersdorff, Litim , Berges, Morris , Graeter, v.Gersdorff, Litim , Berges, Mouhanna, Delamotte, Canet, Bervilliers,Mouhanna, Delamotte, Canet, Bervilliers,
Hubbard model Hubbard model Baier,Bick,…, Metzner et al, Baier,Bick,…, Metzner et al, Salmhofer et al, Honerkamp et al, Krahl, Salmhofer et al, Honerkamp et al, Krahl,
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, PietroniDelamotte, Tissier, Canet, Pietroni
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, 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
wide applicationswide applications
ultracold atomsultracold atoms
Feshbach resonances Feshbach resonances Diehl, Gies, Diehl, Gies, Pawlowski ,…, Krippa,Pawlowski ,…, Krippa,
BEC BEC Blaizot, Wschebor, Dupuis, SenguptaBlaizot, Wschebor, Dupuis, Sengupta
unified description of unified description of scalar models for all d scalar models for all d
and Nand N
Flow equation for average Flow equation for average potentialpotential
Scalar field theoryScalar field theory
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 !
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
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 modeldoped Hubbard model
κ
- ln (k/Λ)
Tc
T>Tc
T<Tc
C.Krahl,… macroscopic scale 1 cm
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
convergence and errorsconvergence and errors
for precise results: systematic for precise results: systematic derivative expansion in second order derivative expansion in second order in derivativesin derivatives
includes field dependent wave includes field dependent wave function renormalization Z(function renormalization Z(ρρ))
fourth order : similar resultsfourth order : similar results apparent fast convergence : no series apparent fast convergence : no series
resummationresummation rough error estimate by different rough error estimate by different
cutoffs and truncationscutoffs and truncations
including fermions :including fermions :
no particular problem !no particular problem !
changing degrees of changing degrees of freedomfreedom
Antiferromagnetic Antiferromagnetic order in the order in the
Hubbard modelHubbard model
A functional renormalization A functional renormalization group studygroup study
T.Baier, E.Bick, …
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
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.
Effective potential in mean Effective potential in mean field theoryfield theory
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,…
Rebosonization and the Rebosonization and the mean field ambiguitymean field ambiguity
Bosonic fluctuationsBosonic fluctuations
fermion loops boson loops
mean field theory
RebosonizationRebosonization
adapt bosonization adapt bosonization 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 such that nofour fermion coupling is generated
Evolution with k-dependentfield variables
Rebosonisation
……cures mean field cures mean field ambiguityambiguity
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
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
running couplingsrunning couplings
unrenormalized mass term
Running mass termRunning mass term
four-fermion interaction ~ m-2 diverges
-ln(k/t)
dimensionless quantitiesdimensionless quantities
renormalized antiferromagnetic order parameter κ
evolution of potential evolution of potential minimumminimum
-ln(k/t)
U/t = 3 , T/t = 0.15
κ
10 -2 λ
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
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 behavior
for 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 !
critical behavior for order critical behavior for order parameter and correlation parameter and correlation
functionfunction
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 !!
end