functional renormalization group in fermionic...

Functional Renormalization Groupin fermionic systems

A. Jakovac

Dept. of Atomic PhysicsEotvos Lorand University Budapest

8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 1 / 36

Outlines

1 Introduction: FRG and fermions

2 LPA approximation for fermions

3 Fermion systems at finite chemical potential

4 Higgs stability bound and irrelevant operators

5 Conclusions


Outlines





5 Conclusions


FRG: a generic view

The behaviour of a subset of the world is governed by somerelevant concepts

eg. for a free falling body, the concepts are the time, positionof the body, velocity, acceleration; eventually air resistance

laws determine connections of the relevant concepts, eg.equation of motion

experience: to different segments of the world belongsdifferent set of relevant concepts ⇒ scientific disciplineseg. quantum mechanics, chemistry, biology, society, astronomy are

all use different concepts.

Their elevant concepts are mutually irrelevant for others.


Quantum Field Theory representation

in QFT we represent the concepts by fields,the laws by effective action coming from path integral

eW [J] =

∫Dφ e iS[φ]+Jφ, iΓ[Φ] = W [J]− JΦ, Φ =

δW [J]

δJ.

advantage: constructive definition

for S → S + 12

∫ΦRΦ the change in the effective action Γ

δΓ =i

2STr δR(Γ(1,1) + R)−1 =

i

2δRG

Wetterich equation ( J. Polchinski (1984), C. Wetterich (1993), T. Morris (1994) )

⇒ formally a one-loop equation

alternative definition of the QFT

effect of fermions:STr: trace with fermion loops −1 factorwe need left-right derivative in case of fermionic modes.


Regulator

In one-parameter scaling require

if Rk→∞ →∞, no fluctuations are allowed: Γk→∞ = S

if Rk→0 → 0, no correction: Γk→0 = Γ.

⇒ Γk alternates between the classical and effective action.

Some regulator types:

sharp cutoff: Rk(p) =∞Θ(p − k)

⇒ Γk is the perfect actioneW [J] =

∫p<kDΦp e

iΓk [φ]+JΦ

R (p)k

p

8

sharp cutoff

Litim’s regulator

k

k2 2

2

(F.J. Wegner, A. Houghton PRA8 (1973) 401; P. Hasenfratz and F. Niedermayer, NPB414 (1994) 785)

optimized regulator ⇒ momentum regularizationpreg = p [ pΘ(p − k) + kΘ(k − p)] (D. F. Litim, Nucl.Phys. B 631, 128 (2002))

Γk not exactly the perfect action, but similar concept.

CSS regulator: multi-parameter form interpolating betweenchoices (I. Nandori, JHEP 1304, 150 (2013)).


Evaluation of the STr in presence of fermionic background

Computation of the STr:

background Φ bosonic as well as ψ fermionic: Γ[Φ, ψ, ψ]

kernel of quadratic fluctuations ϕ, ξ around them

fermionic background can link ξ to ξ and ϕ!

in Nambu representation Ψ =

(ψψT

)Fermion matrix:

(Γ(1,1)k,ΨΨ)ij =

∂

∂ΨiΓk [Ψ]

∂

∂Ψj=

(Γψψ ΓψψΓψψ Γψψ

)Boson-fermion mixing:

(Γ(1,1)k )ij =

(ΓΨΨ ΓΨΦ

ΓΦΨ ΓΦΦ

)These are hypermatrices


Evaluation of the tracelog

(A.J., I. Kaposvari, A. Patkos, arXiv:1510.05782)

Computation: STr ∂kRk(Γ(2) + Rk)−1 = ∂k STr ln(Γ(2) + Rk)

Representation(ΓΨΨ ΓΨΦ

ΓΦΨ ΓΦΦ

)=

(1 C0 1

)(0 BA 0

)(1 C0 1

)where A, B, C , C comes consistency. The Tr log of the RHS is easyto evaluate (α = ±1 for bosons/fermions)

STr log Γ(1,1)k = αΨ Tr log ΓΨΨ + αΦ Tr log ΓΦΦ + αΦ Tr log(1− Γ−1

ΦΦΓΦΨΓ−1ΨΨΓΨΦ)

Diagrammatically (for 3-point fermion-boson couplings):


Outlines





5 Conclusions


Treatment of the Wetterich equation

With On[ϕ] operator basis: Γk [ϕ] =∑n

gn(k)On[ϕ]

Expanding RHS of FRG eq.

∂kgn(k) = βn(g)

gn(k) curves: running couplings

βn = 0 fixed points

General operator basis is too large. . .

we should include all operators, also the most exotic ones

in practice we need an Ansatz


Local Potential Approximation

Local Potential Approximation (LPA) + wave fnct. renorm. (LPA′)

eg . : ΓLPA =

∫ddx

[Zϕ2

(∂µΦ)2 + Uk(Φ)

]Wetterich equations at finite temperature/chemical potential

∂kRk(p) = 2kΘ(p < k) ⇒ no momentum dependence

G ∼ (1∓ n±(ω)) ⇒ distribution functions

Generic structure of the potential evolution:(Ω−1d = 2(4π)d/2Γ( d

2+ 1))

δRG ⇒ ∂kUk = Ωd−1kd∑i

gin±(ωik ± µ)± 1

ωik

gi multiplicities

ω2ik = k2 + m2

i (Φ) with background dependent masses;in 1-component scalar model ω2

ik = k2 + ∂2ΦU.

if m2i =constant we recover the one loop expression.


Local fermionic potential approximation

Does LPA′ work in the fermionic case?

Γk [Ψ]→∫ddx

[Zk ψ∂/ψ + Uk(Ψx)

].

Problem with this approach

ψ2i (x) = 0 because of the fermionic nature⇒ (ψCψ)n = 0 for large enough n!⇒ Uk(Ψ) is a finite polynomial?

But. . .

after bosonization Φ = ψMψ, we can have Ub(Φ) arbitrarypotential

perturbation theory: Γ(n)(x1, . . . , xn) =⟨Tψ(x1) . . . ψ(xn)

⟩6= 0

even in the local limit

⇒ we should properly define the fermionic potential(K.-I. Aoki, K. Morikawa, J.-I. Sumi, H. Terao and M. Tomoyose, Phys. Rev. D61 (2000) 045008)

(AJ, A. Patkos, Phys.Rev. D88 (2013) 065008)


Understanding fermionic LPA

Assumption behind the LPA

propagators vary in spacetime much slower than vertices

evolution from full Γ

n1x = x = ... x2 n1 2y = y = ... y

LPA=⇒

local approximation

yx

numerical values of the diagrams are close

vertex of the 1st diagram: Γ(n)k (x1, . . . , xn)Ψ(x1) . . .Ψ(x2n)

vertex of the 2nd diagram comes from a limiting process:

U(n)k lim

∆V→0

(1

∆V

∫∆V

ψ(x)ψ(x)

)nnotation−→ U

(n)k (ψ(x)ψ(x))n

heuristically: position is not a point, but a patch(k sets the resolution/compositeness scale)


Gross-Neveu model with fermionic LPA

(AJ., A. Patkos, P. Posfay, Eur.Phys.J. C75 (2015) 2, arXiv:1406.3195 [hep-th])

Gross-Neveu model: matter content ψi , i = 1 . . .Nf

S [Ψ] =∫ddx

[Nf∑i=1

ψi∂/ψi +g

2NfI

]where I =

(Nf∑i=1

ψiψi

)2

.

chiral symmetry: ψ → −γ5ψ, ψ → ψγ5

O(Nf ) flavour symmetry

Γk [Ψ] must depend on invariant fermion bilinears

Ansatz for the effective action: take just the original invariant I

Γk [Ψ] =∫ddx

[Zk

Nf∑i=1

ψi∂/ψi + U(I )

]could depend on other invariants (eg. (ψγµψ)2)

it is self-consistent to assume just I dependence!


Computation overview

General structure of the expressions: Γ(2) ∼ G−10 −#Ψ⊗ΨT

⇒ inverse, tracelog can be computed

no flavour mixing ⇒ use background ψ = (ζ, . . . , ζ)

use static background

considerable cancellations occur

Evaluating the integrals using Litim’s regulator we find

∂kUk = kd+1Qd

[4Nf + 1

Z 2k2 + 4IU ′2− 1

Z 2k2 + 4IU ′(U ′ + U)

]where Q−1

d = (4π)d/2Γ( d2 + 1), U = 2U ′ + 4IU ′′.

Fermionic & bosonic type terms! (without explicit bosons)


Flow and fixed points

For 2 < d < 4: one nontrivial fixed point for any Nf :Flow pattern d = 3

critical exponents (agree with bosonized version for Nf =∞)

Θn = d − 2n

(1 +

(n − 1)(n − 2)

2Nf − 1

)for d = 3 only Θn=1 = 1 is relevant.

d = 2 only Gaussian fixed pointd = 4 non-Gaussian FP→∞ for Nf =∞.


Fixed point potential

nth order potential: noconvergence for larger x values.

exact asymptotics can beobtained: y∗as ∼ x

d2(d−1)

⇒ Pade resummation

Physical regime x > 0

⇒ U < 0 as we expected.

physical point is at Ψ = 0.Nf = 2 potentials

Resummation:

y∗(x) = (1 + x2)d

4(d−1) limN→∞

PadeNN

[∑2Nn=1

1n `∗nx

n

(1 + x2)d

4(d−1)

],

PadeNN : resum polynomials with degree 2N to ratio of polynomials with

degree N.


Outlines





5 Conclusions


Finite chemical potential

( G.G. Barnafoldi, A. Jakovac, P. Posfay, arXiv:1604.01717 [hep-th] )

Consider a Yukawa-type fermionic model, with the Ansatz

Γ[ϕ,ψ] =

∫d4x

[ψ(i /∂ − gϕ)ψ +

1

2(∂µϕ)2 − U(ϕ)

].

The FRG equations (ω2B = k2 + ∂2

ϕU, ω2F = k2 + g2ϕ2.)

∂kUk =k4

12π2

[1 + 2nB(ωB)

ωB+ 4−1 + nF (ωF − µ) + nF (ωF + µ)

ωF

]At finite T , µ it is a regular differential equation(T. K. Herbst, J. M. Pawlowski and B. J. Schaefer, PRD88 014007 (2013))

( M. Drews, T. Hell, B. Klein and W. Weise, PRD88 096011 (2013))


T=0 equation

At zero temperature

∂kUk =k4

12π2

[1

ωB− 4

Θ(ωF − µ)

ωF

]

gφ

Λ

k

µ

µ

D

D

<

>SF

Fermi surface is atωF = µ ⇒ k2 + g2ϕ2 = µ2

an ellipse.

Direct polynomial expansion is impossible because of the jump innF (ωF − µ) for small T .Instead: Solve the equations separately in D> and D<, requirecontinuous solution at the Fermi surface.


Solution strategy: the UV domain

D> domain

∂kUk =k4

12π2

[1

ωB− 4

ωF

]the same as for µ = 0 ⇒ find theU> solution using standard methods

gφ

Λ

k

µ

µ

D

D

<

>SF

simplest Ansatz: polynomial expansion to quartic order

U>(k , ϕ) =1

2m2

k +λk24ϕ4,

initial conditions m20, λ0 can be determined from computing

physical observables.

value of the potential at the Fermi surface ϕ = ϕF (k)

U<(k , ϕF (k)) = U>(k , ϕF (k)) ≡ V (k).

⇒ provides boundary conditions for D< domain


Solution strategy: the IR domain

D< domain

∂kUk =k4

12π2

1

ωB

& boundary conditions on Fermi surface

transform boundary to a rectangle bychanging k → x(k), ϕ→ y(k , ϕ)

(evolution should contain first order x derivatives

⇒ x(k) must not depend on ϕ).

choice x = ϕF (k), y =ϕ

x

D>

D<

SF

gφ

k

µ

µ

The new differential equation (u = u − V0)

x∂x u = −xV ′0 + y∂y u −g2(kx)3

12π2

1√(kx)2 + ∂2

y u,

with the boundary conditions u(x = 0, y) = u(x , y = ±1) = 0.


Solution strategy: function basis

To respect boundary conditions expand in a basis hn(y):

u(x , y) =N→∞∑n=1

cn(x)hn(y)

boundary conditions hn(±1) = 0

orthonormal1∫

0

dy hn(y)hm(y) = δnm

possible choice:

hn(y) =√

2 cos((2n + 1)πy2 )

We transform the partial differential equation into an ordinaryintegro-differential equation

∂x u = F(x , ∂py u) ⇒ ∂xcn =1∫

0

dy F(x ,∑

cm∂py hm)hn(y)


Solution strategy: function basis

To perform the integrals analytically

1√(kx)2 + ∂2

y u=

P→∞∑p=0

(−1/2

p

)(∂2

y u −M2)p

((kx)2 + M2)p+1/2

with an appropriate M2 ⇒ optimization.the value of P controls the “loop order”:

P = −1 ⇒ mean field, no bosonic fluctuations

P = 0 ⇒ one-loop with mass M2

P > 0 ⇒ running couplings


Parameters

for the coupling of the Lagrangian we choose “nuclear-type”parameters with large scalar mass

v = fπ = 93 MeV , mN = gv = 938 MeV , m2σ = λv2

3 = mN .

Experience: N ∼ 10− 12 is enough

we varied P = −1, . . . 3

M2 was chosen to have fastest (apparent) convergence


Results: effective potential

convergence as increasing loop (P) parameter

0.2 0.4 0.6 0.8 1.0 1.2Φ fΠ

0.5

1.0

1.5

2.0

2.5

3.0

U

0.2 0.4 0.6 0.8 1.0 1.2Φ fΠ

-1.0

-0.5

0.5

1.0

1.5

2.0U

orange dashed: mean field, up to down: P = 0 . . . 3

left panel: first order transition point for mean field µMF

right panel: first order transition point for FRG µc = 1.053µMF

MF, one-loop are different

for P > 0: good convergence where the curvature is positive

physical U is convex, Maxwell construction between theminima ⇒ for P →∞ a straight line


Results: phase diagram

(λ, g) space: either first order or second order phase transition;border line tricritical points.

0 100 200 300 400 5000

2

4

6

8

10

12

Λ

gcHΛL

2nd order

1st order

HHHHHY

mean field

HHHHHY

one loop

XXXy FRG

mean field predicts strongest phase transition∃ (λ, g) where mean field predicts 1st order, FRG 2nd order PhT.

bosonic fluctuations soften the transition

one-loop is quite good


Results: Equation of State

(G.G.Barnafoldi, AJ., P. Posfay, in preparation)

We can calculate the p(µ) curves in different approximations

from here thermodynamics tell us the p(ε) relation (EoS)

bosonic fluctutions make the EoS (somewhat) stiffer.10% stiffer than mean field, 25% stiffer than 1-loop.

pressure vs µ

EoS (p vs ε)


Results: stability of compact stars

EoS can be used to calculate the compact star stability curvesusing TOV equation. The result is the M(R) relation.

mass-radius diagram

realistic M(R) curves from this simple model, similar to SQMcurves ⇒ improvement is needed

quantum fluctuations tend to stabilize compact starsca. 5-10% effect in this model.


Outlines





5 Conclusions


The Higgs-top system

(AJ., I. Kaposvari, A. Patkos, arXiv:1510.05782 [hep-th]) ⇒ cf. poster of I. KaposvariStandard Model Higgs-sector

important: scalar self-coupling and top-Higgs Yukawa coupling

Fields: σ Higgs field, ψ top field

Action is function of invariants: ρ = 12σ

2, I = σψψ.

FRG Ansatz for this system:

Γk =

∫x

[Zψk ψγm∂mψ +

1

2Zσk(∂mσ)2 + hk I + Uk(ρ)

].

FRG equation for the potential:

∂kU = −1

2Tr ln Γ

(2)ΨΨ +

1

2Tr ln Γ(2)

σσ +1

2Tr ln

(1− Γ(2)−1

σσ Γ(2)σΨΓ

(2)−1ΨΨ Γ

(2)Ψσ

)anomalous dimensions −∂k lnZ = η functional of U

η → 0 in IR limit (irrelevant)


FRG equations, and the running of the Yukawa coupling

∂kU = −1

2Tr ln Γ

(2)ΨΨ +

1

2Tr ln Γ(2)

σσ +1

2Tr ln

(1− Γ(2)−1

σσ Γ(2)σΨΓ

(2)−1ΨΨ Γ

(2)Ψσ

)calculate the RHS (cf. poster of I. Kaposvari)

power expand wrt. the invariants

running of hk Yukawa coupling: expand Γk [I ] to linear order

RHS [I ] = RHS [Is ] + (I − Is)RHS ′[Is ] + . . .

expansion point?

first guess: expand around I = 0(J. Braun, H. Gies and D.D. Scherer, Phys. Rev. D83:085012 (2011))

version A: around the Higgs EoM Is(σ) to zeroth order(ie. the Yukawa coupling does not run)

version B: around the Higgs EoM Is(σ) to first order


FRG running, range of stability

Two effects restrict the FRG evolution:

scalar self-coupling drifts running toward Landau pole:λ(ΛLP)→∞ ⇒ upper bound for mH .

top-Higgs Yukawa coupling drifts running towards instability:λ(Λinst)→ 0 ⇒ lower bound for mH .

for a given mH we have maximal cutoff

104 105 106 107 108

100

200

300

400

500

600

700

800

Λ

Gev

mσ

Gev version A

version B

Ref.[7]

104 105 106 107 10860

80

100

120

140

160

Λ

Gev

mσ

Gev

version A

version B

Ref.[7]

Experimental Higgs mass

small difference between results for different expansion⇒ choice of the expansion point is irrelevant


Background dependent Yukawa coupling

choose h(%)I = (h0 + h1%+ . . . )I

we expect h1 irrelevant: start evolution with λ = 0 at k = Λ

h1 does not contribute to the running soon

-4 -3 -2 -1 0 10.00

0.05

0.10

0.15

0.20

t

λ2

ln k/Λ

λ

h1 = 0

h1 6= 0

104 105 106 107 108 10960

80

100

120

140

Λ

Gev

mH

Gev

Gies, h(ϱ)=h0

Is from EoM, h(ϱ)=h0

Gies, h(ϱ)=h0+h1ϱ

Is from EoM, h(ϱ)=h0+h1ϱ

Gies, h(ϱ)=h0+h1ϱ, for positive t

Is from EoM, h(ϱ)=h0+h1ϱ, for positive t

Λ(GeV)

mH (GeV)

HHj

variation around the original cutoff

HHYeffect of back bending running

But this operator can have important effect towards UV:instead of λ < 0 (unstable system), λ turns back.

Stability bound changes significantly(AJ., I. Kaposvari, A. Patkos, arXiv:1510.05782 [hep-th])

Other irrelevant operators (eg. gravity) can do similar job( M. Shaposhnikov, Ch. Wetterich, Phys.Lett. B683 (2010) 196-200)


Outlines





5 Conclusions


Conclusions

LPA for fermionic systems can be defined ⇒ potentialdepends on (invariant) fermion bilinears to any power

problem at T = 0, µ > 0: presence of a sharp Fermi-surfacesolution: solve FRG equations separately in the high and lowenergy domains, match the solutions on the boundarybosonic fluctuations weaken the phase transitionbut make EoS stiffer, stabilize neutron stars

Higgs stability: position of the maximal cutoff may depend onIR irrelevant operators, like background dependent Yukawacoupling


functional renormalization group in fermionic...

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