Functional renormalization group equation for strongly correlated fermions

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Functional renormalization group equation for
strongly correlated fermions
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collective degrees of freedom
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Hubbard model

• Electrons on a cubic lattice
• here on planes ( d 2 )
• Repulsive local interaction if two electrons are
  on the same site
• Hopping interaction between two neighboring sites

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In solid state physics model for everything

• Antiferromagnetism
• High Tc superconductivity
• Metal-insulator transition
• Ferromagnetism

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Hubbard model
Functional integral formulation
next neighbor interaction
External parameters T temperature µ chemical
potential (doping )
U gt 0 repulsive local interaction
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Fermi surface
Fermion quadratic term
?F (2n1)pT
Fermi surface zeros of P for T0
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Antiferromagnetism in d2 Hubbard model
U/t 3
antiferro- magnetic order parameter
Tc/t 0.115
temperature in units of t
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Collective degrees of freedom are crucial !

• for T lt Tc
• nonvanishing order parameter
• gap for fermions
• low energy excitations
• antiferromagnetic spin waves

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QCD Short and long distance degrees of freedom
are different ! Short distances quarks
and gluons Long distances baryons and
mesons How to make the transition?
confinement/chiral symmetry breaking
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Nambu Jona-Lasinio model
and more general quark meson models
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Chiral condensate (Nf2)
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Functional Renormalization Group

• from small to large scales

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How to come from quarks and gluons to baryons and
mesons ?How to come from electrons to spin waves
?

• Find effective description where relevant
  degrees of freedom depend on momentum scale or
  resolution in space.
• Microscope with variable resolution
• High resolution , small piece of volume
• quarks and gluons
• Low resolution, large volume hadrons

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Effective potential includes all fluctuations
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Scalar field theory

• linear sigma-model for
• chiral symmetry breaking in QCD
• or
• scalar model for antiferromagnetic spin waves
• (linear O(3) model )
• fermions will
  be added later

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Scalar field theory
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Flow equation for average potential
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Simple one loop structure nevertheless (almost)
exact
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Infrared cutoff
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Partial differential equation for function U(k,f)
depending on two ( or more ) variables
Z k c k-?
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Regularisation

• For suitable Rk
• Momentum integral is ultraviolet and infrared
  finite
• Numerical integration possible
• Flow equation defines a regularization scheme (
  ERGE regularization )

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Integration by momentum shells

• Momentum integral
• is dominated by
• q2 k2 .
• Flow only sensitive to
• physics at scale k

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Wave function renormalization and anomalous
dimension

• for Zk (f,q2) flow equation is exact !

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Effective average actionand exact
renormalization group equation
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Generating functional
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Effective average action
Loop expansion perturbation theory
with infrared cutoff in propagator
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Quantum effective action
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Exact renormalization group equation
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Exact flow equation for effective potential

• Evaluate exact flow equation for homogeneous
  field f .
• R.h.s. involves exact propagator in homogeneous
  background field f.

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Flow of effective potential

• Ising model

CO2
Critical exponents
Experiment
T 304.15 K p 73.8.bar ? 0.442 g cm-2
S.Seide
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Antiferromagnetic order in the Hubbard model

• A functional renormalization group study

T.Baier, E.Bick,
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Temperature dependence of antiferromagnetic order
parameter
U 3
antiferro- magnetic order parameter
Tc/t 0.115
temperature in units of t
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Mermin-Wagner theorem ?

• No spontaneous symmetry breaking
• of continuous symmetry in d2 !

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Fermion bilinears
Introduce sources for bilinears Functional
variation with respect to sources J yields
expectation values and correlation functions
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Partial Bosonisation

• collective bosonic variables for fermion
  bilinears
• insert identity in functional integral
• ( Hubbard-Stratonovich transformation )
• replace four fermion interaction by equivalent
  bosonic interaction ( e.g. mass and Yukawa terms)
• problem decomposition of fermion interaction
  into bilinears not unique ( Grassmann variables)

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Partially bosonised functional integral
Bosonic integration is Gaussian or solve
bosonic field equation as functional of fermion
fields and reinsert into action
equivalent to fermionic functional integral if
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fermion boson action
fermion kinetic term
boson quadratic term (classical propagator )
Yukawa coupling
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source term
is now linear in the bosonic fields
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Mean Field Theory (MFT)
Evaluate Gaussian fermionic integral in
background of bosonic field , e.g.
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Effective potential in mean field theory
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Mean field phase diagram
Tc
Tc
µ
µ
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Mean field inverse propagatorfor spin waves
T/t 0.5
T/t 0.15
Pm(q)
Pm(q)
Baier,Bick,
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Mean field ambiguity
Artefact of approximation cured by inclusion
of bosonic fluctuations J.Jaeckel,
Tc
Um U? U/2
U m U/3 ,U? 0
µ
mean field phase diagram
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Flow equationfor theHubbard model
T.Baier , E.Bick ,
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Truncation
Concentrate on antiferromagnetism
Potential U depends only on a a2
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scale evolution of effective potential for
antiferromagnetic order parameter
boson contribution
fermion contribution
effective masses depend on a !
gap for fermions a
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running couplings
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Running mass term
unrenormalized mass term
-ln(k/t)
four-fermion interaction m-2 diverges
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dimensionless quantities
renormalized antiferromagnetic order parameter ?
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evolution of potential minimum
?
10 -2 ?
-ln(k/t)
U/t 3 , T/t 0.15
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Critical temperature
For TltTc ? remains positive for k/t gt 10-9
size of probe gt 1 cm
?
T/t0.05
T/t0.1
Tc0.115
-ln(k/t)
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Pseudocritical temperature Tpc

• Limiting temperature at which bosonic mass term
  vanishes ( ? becomes nonvanishing )
• It corresponds to a diverging four-fermion
  coupling
• This is the critical temperature computed in
  MFT !

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Pseudocritical temperature
Tpc
MFT(HF)
Flow eq.
Tc
µ
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critical behavior
for interval Tc lt T lt Tpc evolution as for
classical Heisenberg model cf.
Chakravarty,Halperin,Nelson
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critical correlation length

c,ß slowly varying functions exponential
growth of correlation length compatible with
observation ! at Tc correlation length reaches
sample size !
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critical behavior for order parameter and
correlation function
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Bosonic fluctuations
boson loops
fermion loops
mean field theory
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Rebosonisation

• adapt bosonisation to every scale k such that
• is translated to bosonic interaction

k-dependent field redefinition
H.Gies ,
absorbs four-fermion coupling
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Modification of evolution of couplings
Evolution with k-dependent field variables
Rebosonisation
Choose ak such that no four fermion coupling is
generated
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cures mean field ambiguity
Tc
MFT
HF/SD
Flow eq.
U?/t
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Nambu Jona-Lasinio model
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Critical temperature , Nf 2
Lattice simulation
J.Berges,D.Jungnickel,
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Chiral condensate
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temperature dependent
masses

• pion mass
• sigma mass

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Criticalequationofstate
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Scalingformofequationof state
Berges, Tetradis,
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Universal critical equation of state is valid
near critical temperature if the only light
degrees of freedom are pions sigma with O(4)
symmetry. Not necessarily valid in QCD, even for
two flavors !
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