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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