Functional renormalization group equation for strongly correlated fermions

Functional Functional renormalization renormalization group equation group equation

for strongly for strongly correlated correlated fermionsfermions

collective collective degrees of degrees of freedomfreedom

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

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

Hubbard modelHubbard model

Functional integral formulation

U > 0 : repulsive local interaction

next neighbor interaction

External parametersT : temperatureμ : chemical potential (doping )

Fermi surfaceFermi surface

Fermion quadratic term

ωF = (2n+1)πT

Fermi surface : zeros of P for T=0

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

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

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

Nambu Jona-Lasinio modelNambu Jona-Lasinio model

……and more general quark meson modelsand more general quark meson models

Chiral condensate Chiral condensate (N(Nff=2)=2)

Functional Renormalization Functional Renormalization GroupGroup

from small to large scalesfrom small to large scales

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

/

Effective potential includes Effective potential includes allall fluctuations fluctuations

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

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

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

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 )

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

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 !

Effective average actionEffective average action

and and

exact renormalization group exact renormalization group equationequation

Generating functionalGenerating functional

Loop expansion :perturbation theory withinfrared cutoffin propagator

Effective average actionEffective average action

Quantum effective actionQuantum effective action

Exact renormalization Exact renormalization group equationgroup equation

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..

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

AntiferromagneticAntiferromagnetic order in the order in the

Hubbard modelHubbard model

A functional renormalization A functional renormalization group studygroup study

T.Baier, E.Bick, …

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

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 !!

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

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,…

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,…

Flow equationFlow equationfor thefor the

Hubbard modelHubbard model

T.Baier , E.Bick , …

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

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 !

Pseudocritical Pseudocritical temperaturetemperature

Tpc

μ

Tc

MFT(HF)

Flow eq.

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

Bosonic fluctuationsBosonic fluctuations

fermion loops boson loops

mean field theory

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

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

Nambu Jona-Lasinio modelNambu Jona-Lasinio model

Critical temperature , NCritical temperature , Nf f = 2= 2

J.Berges,D.Jungnickel,…

Lattice simulation

Chiral condensateChiral condensate

temperature temperature dependent dependent

massesmasses pion masspion mass

sigma masssigma mass

CriticalCriticalequationequation

ofofstatestate

ScalingScalingformform

ofofequationequationof stateof state

Berges,Tetradis,…

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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