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 !!

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