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Peter Kopietz, Frankfurt / sabbatical Gainesville Aug 2013-Feb 2014
The Functional Renormalization Group
1.) Historical introduction: what is the RG?
2.) The basic idea of the Wilsonian RG
3.) Modern formulation: functional RG
4.) Applications
Condensed Matter Seminar, UF Gainesville
13. January 2014
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1.Historical introduction:
What is the renormalization group?
“... the renormalization group is merely a framework, a set of ideas, which has to be adapted to the problem at hand...
All renormalization group studies have in common the idea of re-expressing the parameters which define a problem in terms of some other, perhaps simpler set, while keeping unchanged thosephysical aspects of a problem which are of interest.”(J. Cardy, 1996)
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origin of renormalization: quantum field theory, 1940s
● problem: perturbation theory in quantum electrodynamics gives rise to infinite terms:
● solution: (Bethe, Feynman, Schwinger, Dyson, 1940s)
● all infinities can be absorbed in redefinition (=renormalization) of a finite number of parameters (masses, coupling constants)
● physical quantities can be expressed in terms of finite renormalized couplings (bare couplings are infinite)
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origin of the RG:Stueckelberg and Petermann, 1951
● understand high-energy behavior of renormalized QED
● arbitrariness in definition of renormalized couplings can be used to relate physical correlation functions at different energies
● 1951 paper remained unnoticed, even by QFT experts
● 1953 paper: finite renormalization transformations form a Lie group, for which differential equations hold. (almost unnoticed, because in French)
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RG in quantum field theory: 1950s
● Gell-Mann and Low,1954
● Bogoliubov and Shirkov, 1955
first appearance of the name renormalization group
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Kenneth Wilson, 1970s
● problem in statistical physics: universality of critical exponents:
● new formulation of the RG idea “Wilsonian RG” (more general than field theoretical RG)
● Nobel Prize in Physics 1982: “...for his theory of critical phenomena in connection with phase transitions...”
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Wilsonian RG: pioneering works
● Wilson, Phys. Rev. 1971
● Wilson, Fisher, PRL 1972
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critical exponents for some universality classes
values of critical exponents cannot be obtained from dimensional analysis!
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Literature: Wilsonian RG
● S. K. Ma, 1976:● N. Goldenfeld, 1992:
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Functional RG history
● idea: derive exact functional differential equation describing Wilsonian mode elimination (Wegner and Houghton, 1972)
● most convenient formulation: “Wetterich equation” (Wetterich, 1993)
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2. Wilsonian RG: the basic idea
● explain concepts for Ising model on D-dimensional lattice, nearest neighbor coupling J, magnetic field h:
● want: partition function:
● exact results only in D=1, and D=2 for h=0 (Onsager, 1944)
● first try: mean-field approximation:
● mean-field critical exponents wrong for D < 4
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Ginzburg-Landau-Wilson action
● effective field theory for Ising model: theory:
● bare couplings:
● ultraviolet cutoff:
● partition function becomes functional integral:
● derivation: multi-dimensional Gaussian integral
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Wilson’s iterative RG procedure 1
● Strategy: perform integration iteratively in small steps:
● Step1: Mode elimination (decimation): Integrate over fields describing short wavelength or high energy fluctuations.
carry out integration perturbatively
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Wilson’s iterative RG procedure 2
● effect of mode elimination: modified couplingsto leading (one-loop order):
● Step2: Rescaling:rescale wavevectors and fields such that action after mode elimination has same form as before:
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Wilson’s iterative RG procedure 3
● effect of mode elimination+rescaling:
renormalized couplings:
● iteration in infinitesimal steps: differential equations:
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RG flow close to fixed point
● RG trajectory remains for long time in vicinity of fixed point● microscopic origin of universality
Ginzburg scale
inverse correlationlength
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RG fixed points and critical exponents
● RG fixed points describe scale-invariant system
● critical fixed points:● two relevant directions● infinite correlation length● critical manifold describes
system at critical point
● critical exponents:● are determined by eigenvalues of
linearized RG flow in vicinity of critical fixed points● origin of universality
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3. Functional renormalization group
● main idea: Wilsonian mode elimination can be expressed in terms of formally exact functional differential
equation for generating functionals
● generating functional of Green functions
Green functions:
generating functional:
example: two-point function of Ising model at critical point:
anomalous dimension in D=3.
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FRG flow of generating functionals
● strategy: derive RG equation for generating functionals
information for RG flow of all Green functions
● introduce cutoff: modify Gaussian propagator
● take derivative of generating functional with respect to cutoff FRG flow equation
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exact FRG flow equations
● different types of Green/vertex functions● connected Green functions
(Wegner-Houghton equation, 1972)● amputated connected Green functions
(Polchinski equation, 1984)
technical complication: delta-function*step function:
● one-particle irreducible vertices(Wetterich equation, 1993)
cutoff function second functional derivative
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vertex expansion 1
● functional Taylor expansion
Wetterich equation reduces to infinite hierarchy of integro-differential equations for irreducible vertices
● exact flow equation for irreducible self-energy
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vertex expansion 2
● exact flow equation for effective interaction
three-bodyrenormalizes two-bodyinteraction
● needed: controlled truncation strategies!
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4. Applications: a) Instabilities of normal fermions
(Metzner, Salmhofer, Honerkamp, 2000-today, recent review: Metzner et al. Rev. Mod. Phys. 2012)
● unbiased method for detecting leading instability of normal fermions at weak to intermediate coupling
● numerical implementation needed
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Multi-channel patching trunction
(Halboth, Metzner, PRB 2000; Metzner et al, RMP 2012)
Hubbard model close to half filling:
projection onto Fermi surface:
RG flow of interactions:
RG flow of susceptibilities:
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Applications: b) interacting bosons:
●Bogoliubov-shift:
●Bogoliubov mean-field Hamiltonian:
●condensate density:
●excitation energy:
●long wavelength excitations: sound!
Bogoliubov 1947
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beyond mean-field: infrared divergencies
●Bogoliubov mean-field Hamiltonian:
normal self-energy:
anomalous self-energy:
●mean-field fails due to infrared divergent fluctuation corrections:
for Ginzburg scale
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exact result:Nepomnyashi-identity (1975)
●anomalous self-energy vanishes at zero momentum/frequency:
●Bogoliubov approximation wrong:
●need non-perturbative methods!
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origin of infrared divergencies
●reason for IR divergence: coupling between transverse and longitudinal fluctuations and resulting divergence of longitudinal susceptibility
(Patashinskii+Pokrovskii, JETP 1973)
●consequence: critical continuum in longitudinal part of spectral function
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spectral function and quasi-particle damping in 2D from FRG
●spectral function: ●quasi-particle damping:
(Sinner, Hasselmann, PK, PRL 2007)
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Application: c) strong coupling regime of interacting fermions
● partial bosonization (Wetterich, Gies 2004; PK, Bartosch, Schütz, 2005)
● closing vertex expansion via Ward identities (Streib, Isidori, PK, 2013)
● BCS-BEC crossover (PK, Bartosch, Ferraz 2009)
● non-analytic corrections to Fermi liquid theory (work in progress, 2014)
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Partially bosonized FRG
● Anderson impurity model:
● Hubbard-Stratonovich transformation (partial bosonization):
● exact FRG flow equation for self-energy:
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Closing FRG hierarchy via Ward identities
● Koyama-Tachiki (1984):
● Yamada-Yoshida (1975):
● Dyson-Schwinger equations:
● Friedel sum rule:
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Kondo regime of Anderson impurity model: FRG versus NRG and BA
quasi-particle residue,
spin susceptibiliy:
magnetization curve at strong coupling:
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5.Conclusions, outlook
● FRG gives exact flow equations for all correlation functions
● since 2000: many applications to condensed
matter systems, fermions and bosons
● for strong coupling physics: non-perturbative truncation strategies (e.g. Ward identities)
● generalization to non-equilibrium available
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6.BEC-BCS crossover
● electron gas with attractive interaction:crossover from weakly coupled Cooper pairs to strongly bound fermion pairs:
● qualitative phase diagram: mean field theory ok (Eagles, 1969)
● quantitative calculations in crossover regime difficult
● experimentally accessible with ultracold atoms
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BCS-BEC crossover: mean-field results
● model: fermions with short range two-body attraction
● regularized BCS gap equation:
● mean-field equation for chemical potential:
● dimensionless coupling:
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mean-field results
● gap: ● chemical
potential:
● at unitary point ( ):
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beyond mean-field: FRG flow equations
● truncated FRG flow equations for self-energies:(Bartosch, P.K., Ferraz, PRB, 2009)
● flow of order parameter:
● close system of flow equations using Ward identities and skeleton equations
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truncation of FRG flow equations
superfluid order parameter interaction
vertexsingle-particlegap
order parametercorrelation function
interaction vertex
fermion propagators
●Ward identity: relation between vertex functions of different order due to symmetry:
●skeleton equation: relation between vertex functions (not implied by symmetry):
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FRG results:
● wave-function renormalization, vertex correction
● gap, order parameter, chemical potential
● at unitary point:
agrees with Bartenstein et al (PRL 2004)
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Current project: non-analytic corrections to Fermi liquid theory
● second order perturbation theory:
free energy:
self-energy:
(Maslov, Chubukov, PRB 2009)
● need: non-perturbative calculation of coefficients of non-analytic terms.
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non-perturbative calculation via FRG
● partial bosonization in spin-singlet particle-hole channel
● FRG flow equations (cutoff only in bosonic sector):
free energy:
self-energy:
● bubble from Dyson-Schwinger equation:
● some numerics to get numbers...
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Literature on FRG