Gross-Neveu-Yukawa models at criticality

Benjamin Knorr

Theoretisch-Physikalisches InstitutFriedrich-Schiller-Universitat Jena

DPG Spring Meeting 2017 BremenPhys. Rev. B94 (2016) 245102

Research Training GroupQuantum and Gravitational Fields

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Introduction Renormalisation Results Summary & Outlook

Gross-Neveu model

classical GN model: asymptotically free fermionic theory in 2dimensionsaction:

SGN =

∫d2x

[ψ/∂ψ +

g2Nf

(ψψ

)2]

Gross, Neveu (1974): spontaneous breaking of chiralsymmetry by bifermion condensate → toy model forconfinement

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality

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Introduction Renormalisation Results Summary & Outlook

Why bother?

-∞ -2 -1 - 34 - 1

2 - 14 - 1

8 0 18

14

12

34 1 2 ∞

36π145

72π145

108π145

144π145

gλ

g

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality

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Introduction Renormalisation Results Summary & Outlook

Why bother?

d = 3: relativistic, effective low energy model for fermions,interesting e.g. in condensed matter (graphene)d = 4: Higgs-top sector of Standard Model, low energydescription of QCD (quark-meson model)can be interpreted as asymptotically safe field theory

Here: focus on d = 3 and Nf = 1, 2 flavours of fermions in4-dimensional reducible representation

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality

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Introduction Renormalisation Results Summary & Outlook

QFT, beta functions and the like

calculating the path integral is very difficult[citation needed]:

Z =

∫DΦ eiS[Φ]/ℏ

study approximations within subspace of all possibleinteractionsthis makes observables depend on only a manageable amountof coupling constantscoupling constants depend on energy scale ⇒ encoded in betafunctionscriticality: correlation length diverges, beta functions vanish

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality

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Introduction Renormalisation Results Summary & Outlook

Example: Strong coupling constant

taken from nobelprize.org, popular information on ’04 Nobel Prize in physics

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality

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Introduction Renormalisation Results Summary & Outlook

Renormalisation

beta functions can be calculated by renormalisation grouptechniques (either perturbatively or nonperturbatively)efficient description by effective action Γ (quantum analogueof classical action)

b

b

S

Γ

β

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality

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Introduction Renormalisation Results Summary & Outlook

Partial bosonisation

SGN =

∫d3x

[ψ/∂ψ +

g2Nf

(ψψ

)2]

employ Hubbard-Stratonovich transformation to get rid of4-fermi term (χ ∼ ψψ):

SpbGN =

∫d3x

(ψ

(/∂ + hχ

)ψ − Nf

2 m2χ2)

Yukawa coupling h with g = h2/m2

use functional RG methods to study this model at criticality inextended approximation

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality

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Introduction Renormalisation Results Summary & Outlook

Approximation

Γ ≈∫

d3x[1

2Zχ(ρ)(∂µχ)2 − V (ρ) + gχ(ρ)χψψ

+12Zψ(ρ)

[ψ(1Nf ⊗ /∂)ψ − (∂µψ)(1Nf ⊗ γµ)ψ

]+ i Jψ(ρ)(∂µρ)ψ(1Nf ⊗ γµ)ψ + X1(ρ)χ(∂µψ)(∂

µψ)

+i2X2(ρ)(∂

µχ)(ψ∂µψ − (∂µψ)ψ

)+ X3(ρ)(∂

2χ)ψψ

+12X4(ρ)(∂

µχ)(ψ(1Nf ⊗ Σµν)∂

νψ − (∂νψ)(1Nf ⊗ Σµν)ψ)

+12(X5(ρ) + 2X ′

3(ρ))(∂µχ)2χψψ

]

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality

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Introduction Renormalisation Results Summary & Outlook

Technicalities

by-hand derivation of beta functions practically impossible,computer algebra neededuse xAct to do the job (tensor algebra package for gravity)still have to solve set of 10 ODEs → pseudo-spectral methods:

f (ρ) ≈Nmax∑i=0

cnTn

(2ρ− ρmaxρmax

)

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality

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Introduction Renormalisation Results Summary & Outlook

Nf =2

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality

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Introduction Renormalisation Results Summary & Outlook

Nf =1

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality

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Introduction Renormalisation Results Summary & Outlook

Summary & Outlook

technical progress: extended approximations can be studiedNf = 2 : results appear to be converged, good agreementNf = 1 : substantial disagreement of different methods,further studies necessaryfuture: study other types of condensates:ϕa =

[ψ(σa ⊗ 14)ψ

]

B. Knorr TPI Uni JenaGross-Neveu-Yukawa models at criticality