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Functional Functional renormalization renormalization group equation group equation for strongly for strongly correlated correlated fermions fermions
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Functional renormalization group equation for strongly correlated fermions

Jan 09, 2016

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Functional renormalization group equation for strongly correlated fermions. collective degrees of freedom. Hubbard model. Electrons on a cubic lattice here : on planes ( d = 2 ) Repulsive local interaction if two electrons are on the same site - PowerPoint PPT Presentation
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Page 1: Functional renormalization group equation for strongly correlated fermions

Functional Functional renormalization renormalization group equation group equation

for strongly for strongly correlated correlated fermionsfermions

Page 2: Functional renormalization group equation for strongly correlated fermions

collective collective degrees of degrees of freedomfreedom

Page 3: Functional renormalization group equation for strongly correlated fermions

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

Page 4: Functional renormalization group equation for strongly correlated fermions

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

Page 5: Functional renormalization group equation for strongly correlated fermions

Hubbard modelHubbard model

Functional integral formulation

U > 0 : repulsive local interaction

next neighbor interaction

External parametersT : temperatureμ : chemical potential (doping )

Page 6: Functional renormalization group equation for strongly correlated fermions

Fermi surfaceFermi surface

Fermion quadratic term

ωF = (2n+1)πT

Fermi surface : zeros of P for T=0

Page 7: Functional renormalization group equation for strongly correlated fermions

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

Page 8: Functional renormalization group equation for strongly correlated fermions

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

Page 9: Functional renormalization group equation for strongly correlated fermions

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

Page 10: Functional renormalization group equation for strongly correlated fermions

Nambu Jona-Lasinio modelNambu Jona-Lasinio model

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

Page 11: Functional renormalization group equation for strongly correlated fermions

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

Page 12: Functional renormalization group equation for strongly correlated fermions

Functional Renormalization Functional Renormalization GroupGroup

from small to large scalesfrom small to large scales

Page 13: Functional renormalization group equation for strongly correlated fermions

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

Page 14: Functional renormalization group equation for strongly correlated fermions
Page 15: Functional renormalization group equation for strongly correlated fermions

/

Page 16: Functional renormalization group equation for strongly correlated fermions

Effective potential includes Effective potential includes allall fluctuations fluctuations

Page 17: Functional renormalization group equation for strongly correlated fermions

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

Page 18: Functional renormalization group equation for strongly correlated fermions

Scalar field theoryScalar field theory

Page 19: Functional renormalization group equation for strongly correlated fermions

Flow equation for average Flow equation for average potentialpotential

Page 20: Functional renormalization group equation for strongly correlated fermions

Simple one loop structure –Simple one loop structure –nevertheless (almost) exactnevertheless (almost) exact

Page 21: Functional renormalization group equation for strongly correlated fermions

Infrared cutoffInfrared cutoff

Page 22: Functional renormalization group equation for strongly correlated fermions

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

Page 23: Functional renormalization group equation for strongly correlated fermions

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 )

Page 24: Functional renormalization group equation for strongly correlated fermions

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

Page 25: Functional renormalization group equation for strongly correlated fermions

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 !

Page 26: Functional renormalization group equation for strongly correlated fermions

Effective average actionEffective average action

and and

exact renormalization group exact renormalization group equationequation

Page 27: Functional renormalization group equation for strongly correlated fermions

Generating functionalGenerating functional

Page 28: Functional renormalization group equation for strongly correlated fermions

Loop expansion :perturbation theory withinfrared cutoffin propagator

Effective average actionEffective average action

Page 29: Functional renormalization group equation for strongly correlated fermions

Quantum effective actionQuantum effective action

Page 30: Functional renormalization group equation for strongly correlated fermions

Exact renormalization Exact renormalization group equationgroup equation

Page 31: Functional renormalization group equation for strongly correlated fermions

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..

Page 32: Functional renormalization group equation for strongly correlated fermions

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

Page 33: Functional renormalization group equation for strongly correlated fermions

AntiferromagneticAntiferromagnetic order in the order in the

Hubbard modelHubbard model

A functional renormalization A functional renormalization group studygroup study

T.Baier, E.Bick, …

Page 34: Functional renormalization group equation for strongly correlated fermions

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

Page 35: Functional renormalization group equation for strongly correlated fermions

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 !!

Page 36: Functional renormalization group equation for strongly correlated fermions

Fermion bilinearsFermion bilinears

Introduce sources for bilinears

Functional variation withrespect to sources Jyields expectation valuesand correlation functions

Page 37: Functional renormalization group equation for strongly correlated fermions

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)

Page 38: Functional renormalization group equation for strongly correlated fermions

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

Page 39: Functional renormalization group equation for strongly correlated fermions

fermion – boson actionfermion – boson action

fermion kinetic term

boson quadratic term (“classical propagator” )

Yukawa coupling

Page 40: Functional renormalization group equation for strongly correlated fermions

source termsource term

is now linear in the bosonic fields

Page 41: Functional renormalization group equation for strongly correlated fermions

Mean Field Theory (MFT)Mean Field Theory (MFT)

Evaluate Gaussian fermionic integralin background of bosonic field , e.g.

Page 42: Functional renormalization group equation for strongly correlated fermions

Effective potential in mean Effective potential in mean field theoryfield theory

Page 43: Functional renormalization group equation for strongly correlated fermions

Mean field phase Mean field phase diagramdiagram

μμ

TcTc

Page 44: Functional renormalization group equation for strongly correlated fermions

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,…

Page 45: Functional renormalization group equation for strongly correlated fermions

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,…

Page 46: Functional renormalization group equation for strongly correlated fermions

Flow equationFlow equationfor thefor the

Hubbard modelHubbard model

T.Baier , E.Bick , …

Page 47: Functional renormalization group equation for strongly correlated fermions

TruncationTruncationConcentrate on antiferromagnetism

Potential U depends only on α = a2

Page 48: Functional renormalization group equation for strongly correlated fermions

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

Page 49: Functional renormalization group equation for strongly correlated fermions

running couplingsrunning couplings

Page 50: Functional renormalization group equation for strongly correlated fermions

unrenormalized mass term

Running mass termRunning mass term

four-fermion interaction ~ m-2 diverges

-ln(k/t)

Page 51: Functional renormalization group equation for strongly correlated fermions

dimensionless quantitiesdimensionless quantities

renormalized antiferromagnetic order parameter κ

Page 52: Functional renormalization group equation for strongly correlated fermions

evolution of potential evolution of potential minimumminimum

-ln(k/t)

U/t = 3 , T/t = 0.15

κ

10 -2 λ

Page 53: Functional renormalization group equation for strongly correlated fermions

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

Page 54: Functional renormalization group equation for strongly correlated fermions

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 !

Page 55: Functional renormalization group equation for strongly correlated fermions

Pseudocritical Pseudocritical temperaturetemperature

Tpc

μ

Tc

MFT(HF)

Flow eq.

Page 56: Functional renormalization group equation for strongly correlated fermions

critical behaviorcritical behavior

for interval Tc < T < Tpc

evolution as for classical Heisenberg model

cf. Chakravarty,Halperin,Nelson

Page 57: Functional renormalization group equation for strongly correlated fermions

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 !

Page 58: Functional renormalization group equation for strongly correlated fermions

critical behavior for order critical behavior for order parameter and correlation parameter and correlation

functionfunction

Page 59: Functional renormalization group equation for strongly correlated fermions

Bosonic fluctuationsBosonic fluctuations

fermion loops boson loops

mean field theory

Page 60: Functional renormalization group equation for strongly correlated fermions

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

Page 61: Functional renormalization group equation for strongly correlated fermions

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

Page 62: Functional renormalization group equation for strongly correlated fermions

……cures mean field cures mean field ambiguityambiguity

Tc

Uρ/t

MFT

Flow eq.

HF/SD

Page 63: Functional renormalization group equation for strongly correlated fermions

Nambu Jona-Lasinio modelNambu Jona-Lasinio model

Page 64: Functional renormalization group equation for strongly correlated fermions

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

J.Berges,D.Jungnickel,…

Lattice simulation

Page 65: Functional renormalization group equation for strongly correlated fermions

Chiral condensateChiral condensate

Page 66: Functional renormalization group equation for strongly correlated fermions

temperature temperature dependent dependent

massesmasses pion masspion mass

sigma masssigma mass

Page 67: Functional renormalization group equation for strongly correlated fermions

CriticalCriticalequationequation

ofofstatestate

Page 68: Functional renormalization group equation for strongly correlated fermions

ScalingScalingformform

ofofequationequationof stateof state

Berges,Tetradis,…

Page 69: Functional renormalization group equation for strongly correlated fermions

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 !

Page 70: Functional renormalization group equation for strongly correlated fermions

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