Functional Renormalization Group and some Applications · 1 K. Aoki, Introduction to the...

Functional Renormalization Group and someApplications

A. Wipf

Theoretisch-Physikalisches Institut, FSU Jena

Humboldt Universität Berlin - NIC,DESY ZeuthenJoint Lattice Seminar

Andreas Wipf Functional Renormalization Group and some Applications

1 Introduction

2 Scale-dependent Functionals

3 Derivation of Flow Equations

4 Functional Renormalization in QM

5 Scalar Field Theories

6 Some applications


Introduction 3 / 45

particular implementation of the renormalization groupfor continuum field theory (symmetries ,)functional methods + renormalization group ideaconceptionally simple, technically demandingscale-dependent Schwinger functional or effective actionlarge values of scale parameter k : high resolution→ UVlowering k : including more and more fluctuations→ IRknown microscopic laws −→ macroscopic phenomena


flow of Schwinger functional Wk [j]: Polchinski equationflow of effective action Γk [ϕ]: Wetterich equationflow from classical action S[ϕ] to effective action Γ[ϕ]

applied to variety of physical systemsstrong interactionelectroweak phase transitionasymptotic safety scenariocondensed matter systene.g. Hubbard model, liquid He4, frustrated magnets, superconductivity . . .effective models in nuclear physicsultra-cold atoms


Γk=Λ = S

Γk=0 = Γ

Theory space


1 K. Aoki, Introduction to the nonperturbative renormalization group and its recentapplications,Int. J. Mod. Phys. B14 (2000) 1249.

2 J. Berges, N. Tetradis and C. Wetterich, Nonperturbative renormalization flow inquantum field theory and statistical physics,Phys. Rept. 363 (2002) 223.

3 H. Gies, Introduction to the functional RG and applications to gauge theories,in Lect. Notes Phys. Volume 852, Springer, Berlin (2012)

4 P. Kopietz, L. Bartosch and F. Schütz, Introduction to the functionalrenormalization group,Lect. Notes in Phys., Volume 798, Springer, Berlin (2010).

5 A. Wipf, Statistical Approach to Quantum Field Theory,Lect. Notes Phys., Volume 864, Springer, Berlin (2013).


Scale-dependent functionals 7 / 45

generating functional of (Euclidean) correlation functions

Z [j] =

∫Dφ e−S[φ]+(j,φ), (j , φ) =

∫ddx j(x)φ(x)

Schwinger functional W [j] = log Z [j]effective action = Legendre transform of W [j]

Γ[ϕ] = (j , ϕ)−W [j] with j(x) from ϕ(x) =δW [j]δj(x)

encodes properties of QFT in most economic way


add scale-dependent IR-cutoff term ∆Sk to S

Zk [j] =

∫Dφ e−S[φ]+(j,φ)−∆Sk [φ]

Scale-dependent Schwinger functional

Wk [j] = log Zk [j]

regulator: quadratic functional

∆Sk [φ] =12

∫ddxddy φ(x)Rk (x − y)φ(y)

→ one-loop structure of flow equation


Conditions on cutoff function Rk (p) 9 / 45

cutoff-function ∼ momentum-dependent mass

∆Sk [φ] =12

∫ddp

(2π)d φ∗(p)Rk (p)φ(p)

recover effective action for k → 0:

Rk (p)k→0−→ 0 for fixed p

recover classical action at UV-scale Λ:

Rkk→Λ−→ ∞

regularization in the IR:

Rk (p) > 0 for p → 0


Cut-offs in use 10 / 45

exponential regulator: Rk (p) =p2

ep2/k2 − 1,

optimized regulator: Rk (p) = (k2 − p2) θ(k2 − p2) ,

quartic regulator: Rk (p) = k4/p2 ,

sharp regulator: Rk (p) =p2

θ (k2 − p2)− p2 ,

Callan-Symanzik regulator: Rk (p) = k2


exponential cutoff function and its derivative

p2

Rk

k∂kRk

k2 2k2

k2

2k2


Polchinski equation 12 / 45

scale dependence of Wk from

∂k Wk [j] = −12

∫ddxddy 〈φ(x)∂k Rk (x , y)φ(y)〉k

relates to connected two-point function

W (2)k (x , y) ≡ δ2Wk [j]

δj(x)δj(y)= 〈φ(x)φ(y)〉k − ϕ(x)ϕ(y)

Polchinski equation

∂k Wk [j] =− 12

∫ddxddy ∂k Rk (x , y) W (2)

k (x , y)

−∫

ddxddy W (1)k ∂k Rk (x , y)W (1)

k (y)


Scale dependent effective action 13 / 45

average field of cutoff theory with j

ϕ(x) =δWk [j]δj(x)

≡W (1)k (x)

fixed j → ϕ depends on kfixed ϕ→ j depends on kmodified Legendre transformation:

Γk [ϕ] =(LWk

)[ϕ]−∆Sk [ϕ]

Legendre transform(LWk

)(ϕ) = (j , ϕ)−Wk [j]

Γk not L-transform of Wk [j] for k > 0!need not be convex, but Γk→0 = Γ convex


Derivation of Wetterich equation 14 / 45

vary effective average action (ϕ fixed)

δΓk [ϕ]

δϕ(x)=

∫δj(y)

δϕ(x)ϕ(y) + j(x)−

∫δWk [j]δj(y)

δj(y)

δϕ(x)− δ∆Sk [ϕ]

δϕ(x)

red terms chancel→ effective equation of motion

δΓk [ϕ]

δϕ(x)= j(x)− δ

δϕ(x)∆Sk [ϕ] = j(x)− (Rkϕ)(x)

flow equation: ϕ fixed, j depends on scale, differentiate Γk

∂k Γk [ϕ] =

∫ddx ∂k j(x)ϕ(x)− ∂k Wk [j]−

∫∂Wk [j]∂j(x)

∂k j(x)− ∂k ∆Sk [ϕ]

PE=

12


k (x , y)


∂k Wk [j] only from k -dependence of the parameters

∂k Γk [ϕ] = −∂k Wk [j]− ∂k ∆Sk [ϕ]

= −∂k Wk [j]− 12

∫ddxddy ϕ(x)∂k Rk (x , y)ϕ(y)

use Polchinski equation→

∂k Γk [ϕ] =12


k (x , y)

second derivative of Wk vs. second derivative of Γk :

ϕ(x) = W (1)k (x) and j(x) = Γ

(1)k (x) +

∫ddy Rk (x , y)ϕ(y)


chain rule→

δ(x − y) =

∫ddz

δϕ(x)

δj(z)

δj(z)

δϕ(y)

=

∫ddz W (2)

k (x , z){

Γ(2)k + Rk

}(z, y)

relation between curvatures

W (2)k =

1

Γ(2)k + Rk

insert into ∂k Γk ⇒

Wetterich equation

∂k Γk [ϕ] =12

tr

(∂k Rk

Γ(2)k [ϕ] + Rk

)


non-linear functional integro-differential equation

full propagator Γ(2)k [ϕ] enters flow equation

Polchinski and Wetterich equations = exact FRG equationsPolchinski: simple polynomial structure (structural investigations)Wetterich: rational structure stabilizes flow (explicit calculations)UV and IR regulatedin practice: truncation = projection onto finite-dim. spaceproblem: reliable error estimate for flow→ improve truncation, vary regulator, check stability


Functional renormalization in QM 18 / 45

anharmonic oscillator

S[q] =

∫dτ(

12

q2 + V (q)

),

here LPA (local potential approximation)

Γk [q] =

∫dτ(

12

q2 + uk (q)

)leading order in gradient expansionscale-dependent effective potential uk

neglected: higher derivative terms, mixed terms qnqm


in denominator of flow equation Γ(2)k = −∂2

τ + u′′k (q)

LPA: may consider constant q → momentum space

(∂k Γk )[q] =∂

∂k

∫dτ uk (q) =

12

∫dτ

∞∫−∞

dp2π

∂k Rk (p)

p2 + u′′k (q) + Rk (p)

optimal regulator

Rk (p) = (k2 − p2) θ(k2 − p2) =⇒ ∂k Rk (p) = 2kθ(k2 − p2)

⇒ non-linear partial differential equation for uk :

∂k uk (q) =1π

k2

k2 + u′′k (q)


free particle limit fixes subtraction in flow equation (Λ→∞)

∂k uk (q) =1π

(k2

k2 + u′′k (q)− 1)

= −1π

u′′k (q)

k2 + u′′k (q)

assume uΛ(q) even→ uk (q) evenpolynomial ansatz

uk (q) = Ek +ω2

k2

q2 +λk

4!q4 +

∑n=3,4,...

1(2n)!

a2n,k q2n ,


insert and compare coefficients→ infinite set of coupled ode’s

dEk

dk= −1

πω2

k ∆0, ∆0 =1

k2 + ω2k,

dω2k

dk= −k2

πλk ∆2

0 ,

dλk

dk= −

k2∆20

π

(a6,k − 6λ2

k ∆0),

da6,k

dk= −

k2∆20

π

(a8,k − 30λk a6,k ∆0 + 90λ3

k ∆20),

......

initial condition: classical potential at cutoffprojection onto space of polynomials up to given degree n


e.g. crude truncation a6 = a8 = · · · = 0truncated finite system of flow equations

dEk

dk= −

ω2kπ

∆0,dω2

kdk

= −k2λk

π∆2

0,dλk

dk=

6k2λ2k

π∆3

0

at cutoff k = Λ: EΛ = 0, ωΛ = 1, varying λΛ at the cutoff scale→ scale-dependent couplingsdominant contribution from near typical scale k ≈ ω


k

Ek

.1

.2

.3

.4

.5

1 2 3 4

from above:

λ = 2.0λ = 1.0λ = 0.5λ = 0.0

k

ω2k

1.0

1.1

1.2

1.3

1.4

1 2 3 4

from above:

λ = 0.0λ = 0.5λ = 1.0λ = 2.0

The flow of the couplings Ek and ω2k (EΛ = 0, ω2

Λ = 1).


ω = ωk=0 > 0 ⇒ minimum of u0 at origin⇒ E0 = u0(0)

energy of first excited state

E1 = E0 +√

u′′0 (0) = E0 + ω0

reasonably good results with 3-parameter truncation

units of ~ω: energies for different λ; different regulators

ground state energy energy of first excited statecutoff optimal optimal Callan exact optimal optimal Callan exact

order 4 order 12 order 4 result order 4 order 12 order 4 resultλ = 0 0.5000 0.5000 0.5000 0.5000 1.5000 1.5000 1.5000 1.5000λ = 1 0.5277 0.5277 0.5276 0.5277 1.6311 1.6315 1.6307 1.6313λ =2 0.5506 0.5507 0.5504 0.5508 1.7324 1.7341 1.7314 1.7335λ = 3 0.5706 0.5708 0.5703 0.5710 1.8177 1.8207 1.8159 1.8197λ = 4 0.5885 0.5889 0.5882 0.5891 1.8923 1.8968 1.8898 1.8955λ = 5 0.6049 0.6054 0.6045 0.6056 1.9593 1.9652 1.9562 1.9637λ = 6 0.6201 0.6207 0.6196 0.6209 2.0205 2.0278 2.0168 2.0260λ = 7 0.6343 0.6350 0.6336 0.6352 2.0771 2.0857 2.0728 2.0836λ = 8 0.6476 0.6484 0.6469 0.6487 2.1299 2.1397 2.1250 2.1374λ = 9 0.6602 0.6611 0.6594 0.6614 2.1794 2.1905 2.1741 2.1879λ = 10 0.6721 0.6732 0.6713 0.6735 2.2263 2.2385 2.2205 2.2357λ = 20 0.7694 0.7714 0.7679 0.7719 2.5994 2.6209 2.5898 2.6166


negative ω2: mexican hat

∂k uk (q) = −1π

u′′k (q)

k2 + u′′k (q)

denominator positive for large scales⇒ remains positive during the flowflow equation⇒uk (q) increases toward infrared if u′′k (q) is positiveuk (q) decreases toward infrared if u′′k (q) is negativek2 + u′′k minimal at q = 0⇒ double-well potential flattens during flow, becomes convexexpected on general grounds


solution of partial differential equation, ω2 = −1, λ = 1

q

uk

v

.5

V

uk=0


increasing barrier (weak coupling)→ increasingly difficultnumerical solution does bettersplitting induced by instanton effects: beyond leading order LPA

units of ~ω: varying λ; optimized regulator,ground state energy energy of first excited state

optimal optimal pde exact optimal optimal pde exactorder 4 order 12 order 4 order 12

λ = 1 -0.8732 -0.8556 -0.7887 -0.8299λ = 2 -0.2474 -0.2479 -0.2422 0.0049 0.0063 -0.0216λ = 3 0.2473 -0.0681 -0.0679 -0.0652 -0.2241 0.3514 0.3500 0.3307λ = 4 -0.0186 0.0286 0.0290 0.0308 0.3511 0.5753 0.5755 0.5598λ = 5 0.0654 0.0949 0.0953 0.0967 0.5835 0.7455 0.7462 0.7324λ = 6 0.1234 0.1457 0.1461 0.1472 0.7509 0.8842 0.8851 0.8723λ = 7 0.1688 0.1871 0.1876 0.1885 0.8851 1.0021 1.0030 0.9909λ = 8 0.2063 0.2223 0.2228 0.2236 0.9987 1.1052 1.1061 1.0944λ = 9 0.2671 0.2530 0.2535 0.2543 1.1863 1.1972 1.1981 1.1866λ = 10 0.2386 0.2803 0.2808 0.2816 1.0978 1.2805 1.2814 1.2701λ = 20 0.4536 0.4632 0.4639 0.4643 1.7866 1.8638 1.8648 1.8538


Scalar Field Theory 28 / 45

LP approximation

Γk [ϕ] =

∫ddx

(12

(∂µϕ)2 + uk (ϕ)

)scale-dependent effective potential

∂k uk (q) =12

∫ddp

(2π)d∂k Rk (p)

p2 + u′′k (q) + Rk (p)

optimized regulator: p-integration doable→ flow equation

∂k uk (ϕ) = µdkd+1

k2 + u′′k (ϕ), µd =

(4π)−d/2

Γ( d2 + 1)

space-time dimensions in kd+1 and µd

nonlinear PDE


series expansion (even)→ flow equations for infinite set of couplings

kda0

dk= +µdkd+2∆0, ∆0 =

1k2 + a2

,

kda2

dk= −µdkd+2∆2

0a4 ,

kda4

dk= −µdkd+2∆2

0(a6 − 6a2

4∆0),

kda6

dk= −µdkd+2∆2

0(a8 − 30a4a6∆0 + 90a3

4∆20),

......


Fixed points 30 / 45

point in space of dimensionless couplings where all β-functions vanish

K2

K1

line of critical points

T < Tc

T > Tc

(K∗1,K∗

2): fixed point

(K1c,K2c): critical point


critical hyper-surface on which ξ =∞RG trajectory moves away from critical surfaceIf flow begins on critical surface→ stays on surfacegenerically: critical points are not fixed pointd ≥ 3 : expect finite set of isolated fixed points K ∗ = (K ∗1 ,K

∗2 , . . . )

RG flow in the vicinity of fixed point K = K ∗ + δKlinearize flow around fixed point

K ′i = K ∗i + δK ′i = Ri(K ∗j + δKj

)= K ∗i +

∂Ri

δKj

∣∣K∗δKj + O(δK 2)


linearized RG transformation,

δK ′i =∑

j

M ji δKj , M j

i =∂Ri

∂Kj

∣∣∣K∗

left-eigenvectors Φαand eigenvalues λα = eyα of Myα > 0: flow repelled by K ∗ → relevant perturbationyα < 0: flow attracted to K ∗ → irrelevant perturbationrel. couplings/exponents→ IR-physics (scaling laws, TD criticalexponents)


Fixed point analysis for scalar models 33 / 45

introduce dimensionless field and potential,

ϕ = k (d−2)/2√µd χ and uk (ϕ) = kdµdvk (χ)

flow equation in terms of dimensionless quantities (rescaled by k#)

k∂k vk + dvk −d − 2

2χv ′k =

11 + v ′′k

, v ′k =∂vk

∂χ. . .

fixed point equation for effective potential (ode)

dv∗ −d − 2

2χv ′∗ =

11 + v ′′∗

constant solution dv∗ = 1→ trivial Gaussian fixed point


scale invariance at fixed point⇒ fixed point ∼ CFT

are there non-Gaussian fixed point solutions?answer depends on spacetime dimension d2d theories:∞ many fixed-point scalar models [min. models; Morris 1994]∞ many fixed-point Yukawa models [Synatschke, AW, ...]3d theories:small number of fixed-pointsfrom various truncations: series expansions and numerical solutionsexample: LPA and polynomial truncation for scalar field theory

w =m∑

n=0

cnρn, ρ =

12χ2


at fixed point: βn(c∗0 , c∗1 , . . . ) = 0 for n = 0,1,2, . . .

always Gaussian fixed point: c∗0 = 13 , 0 = c∗2 = c∗3 = c∗4 = . . .

search for non-trivial fixed pointsonly one for scalar field in d = 3 (even u):m = 20⇒ c∗1 = −.186066m = 42⇒ c∗1 = −.186041⇒ c∗0 , c

∗2 , . . . , c

∗m−1 ⇒ polynomial approximation to fixed point solution

fixed-point coefficients n!c∗n

c∗0 c∗1 c∗2 c∗3 c∗4 c∗5 c∗6m = 20 0.409534 -0.186066 0.082178 0.018981 0.005253 0.001104 -0.000255m = 42 0.409533 -0.186064 0.082177 0.018980 0.005252 0.001104 -0.000256

c∗7 c∗8 c∗9 c∗10 c∗11 c∗12 c∗13m = 20 -0.000526 -0.000263 0.000237 0.000632 0.000438 -0.000779 -0.002583m = 42 -0.000526 -0.000263 0.000236 0.000629 0.000431 -0.000799 -0.002643

c∗14 c∗15 c∗16 c∗17 c∗18 c∗19 c∗20m = 20 -0.002029 0.007305 0.028778 0.034696 -0.077525 -0.381385 0.000000m = 42 -0.002216 0.006677 0.026544 0.026320 -0.110498 -0.517445 -0.587152


Polynomial approximations vs. numerical solution 36 / 45

numerics: shooting method with seventh-order Runge-Kuttaw∗

0 1 2 3 4 5 6 7 8 9 10 11

1

2

3

4

5

6

m = 10

m = 20

m = 30

m = 40

numerical

solution


Critical exponents

m ν = −1/ω1 ω2 ω3 ω4 ω510 0.648617 0.658053 2.985880 7.502130 17.91349414 0.649655 0.652391 3.232549 5.733445 9.32485818 0.649572 0.656475 3.186784 5.853987 9.14109322 0.649554 0.655804 3.170538 5.977066 8.52281126 0.649564 0.655629 3.182910 5.897290 8.84463230 0.649562 0.655791 3.180847 5.903039 8.90760734 0.649561 0.655749 3.178636 5.922910 8.70258338 0.649562 0.655731 3.180577 5.908885 8.81422542 0.649562 0.655755 3.180216 5.909910 8.84738646 0.649562 0.655746 3.179541 5.915754 8.738608

convergencetwo relevant operators: ω0 = −3 and ω1 = −1/νLPA-prediction: ν = 0.649562 (high-T expansion: ν = 0.630)


Wave function renormalization 38 / 45

next-to-leading in derivative expansion→ Zk (p, ϕ)

simple LPA’ truncation (no p and ϕ-dependence)

Γk [ϕ] =

∫ddx

(12

Zk (∂µϕ)2 + uk (ϕ)

).

flow of Zk : project flow on operator (∂φ)2

must admit inhomogeneous ϕ in flow equation⇒ anomalous dimension

η = −k∂k log Zk


Application: linear O(N) models susy: Litim, Synatschke, Heilmann, AW 39 / 45

scalar field φ ∈ RN , O(N) invariant potentialfixed-point analysis: dimensionless field χ and composite field

% =12χ · χ

dimensionless potential νk (%).N − 1 Goldstone modes, one massive radial modeGoldstone modes drive flow for large Nfinds one Wilson-Fisher fixed point w∗ for all Nfixed point analysis: wk = w∗ + δk

fluctuation δk obeys the linear differential equation


polynomial truncation to high order (40)

N 1 2 3 100 1000−w ′∗(0) 0.186064 0.230186 0.263517 0.384172 0.387935ν = −1/ω1 0.64956 0.70821 0.76113 0.99187 0.99923ω2 0.6556 0.6713 0.6990 0.97218 0.99844ω3 3.1798 3.0710 3.0039 2.98292 2.99554

asymptotic formulas, e.g. ν ≈ 0.9998− 0.9616N

Large-N limit

flow equation can be solved (characteristics), exact relation w∗ ⇔ δ in

w(t , ρ) ≈ w∗(ρ) + eωtδ(ρ), δ = w ′ (ω+d)/2∗ t = log

kΛ

perturbation non-singular→ all critical exponents

ωn ∈ {2n − d |n = 0,1,2, . . . }


Special features of susy theories 41 / 45

on-shell supersymmetry: susy transformations non-linear⇒ no supersymmetric quadratic regulator existsoff-shell supersymmetry: susy transformation linearsupersymmetric quadratic regulator constructed Synatschke, Gies, AW

cutoff function bosons⇔ cutoff function fermions

constructed manifest supersymmetric FRG

N = 1 theory in d = 4: non-renormalization of superpotentialN = 1 theories in d < 4: running superpotential, running couplingsN = 2 theories in d < 4: non-renormalization of superpotentialinstead: flow of Kähler-potential


N = 2 Wess-Zumino model in 3 dimensions 42 / 45

Lagrangian density: complex φ,F , Dirac ψ

L = −14

∫d2θd2θΦΦ† −

{ 12i

∫d2θW + h.c.

}= |∇φ|2 + iψσ∇ψ − |F |2 −

{∂W∂φ

F − 12∂2W∂φ2 ψ

Tσ2ψ + h.c.}

(nonlocal) susy regulator with cutoff functions r1(k ,p) and r2(k ,p)

flow of Kähler metric

Γk =

∫d3x(

Z 2k (φ)

{|∇φ|2 + iψσ∇ψ − |F |2

}−{∂Wk

∂φF − 1

2∂2Wk

∂φ2 ψTσ2ψ + h.c.})

⇒ Kähler metric Z 2∗ at fixed point P. Feldmann, L. Zambelli, AW, 2018


Some results on supersymmetric systems 43 / 45

FP-structure in 2 ≤ d ≤ 4non-renormalization theorems with F. Synatschke and M. Heilmann

flow with higher derivative operators with R. Flore, T. Hellwig, B. Knorr and O. Zanusso

exact solution for Wk for susy O(N →∞) model with M. Heilmann and D. Litim

fixed point of nonlinear susy O(N) model with R. Flore and O. Zanusso

spike plots for susy theories with T. Hellwig and O. Zanusso

emergence of supersymmetry in Yukawa models with T. Hellwig and L. Zambelli

flow of Kähler metric in N = 2 in d = 3 with P. Feldmann and L. Zambelli

spontaneouly breaking of susy with H.Gies, F. Synatschke

particle massesfinite-temperature phase transition with J. Braun, F. Synatschke


many FRG studies of Gauge Theories and Gravityproblem with local gauge invariancebackground-field methodbimetric approach, .....

missing: convincing FRG studies for SYMsusy cutoffs exist (beyond WZ gauge)mixing of susy transformations with gauge transformationsneeded for comparison with lattice studiesextract effective theories

supersymmetric lattice gauge theoriesmany lattice-results in d = 1,2 many; D. August, B. Wellegehausen, AW

view lattice simulations in d = 4 Münster-DESY collaboration

M. Steinhauser, A. Sternbeck, B. Wellegehausen, AW


thanks to ... 45 / 45

Coworkers:Jens Braun,Holger Gies, Ilya Shapiro; Boris Merzlikin, Luca Zambelli, Omar Zanusso; Polina Feldmann, Raphael

Flore, Marianne Heilmann, Tobias Hellwig, Franziska Synatschke

JHEP 1712 (2017) 132 JHEP 1502 (2015) 109 JHEP 0903 (2009) 028

Phys. Rev. D96 (2017) 125007; Phys. Rev. D92 (2015) 085027; Phys. Rev. D87 (2013) 06501

Phys. Rev. D86 (2012) 105006; Phys. Rev. D84 (2011) 125009; Phys. Rev. D81 (2010) 125001

Phys. Rev. D80 (2009) 085007; Phys. Rev. D80 (2009) 101701


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Transcript of Functional Renormalization Group and some Applications · 1 K. Aoki, Introduction to the...