FRG ERG E xact RG E xact RG from first principles includes - PowerPoint PPT Presentation

Renormalization flow of relativistic fermions (2<d<4) Holger Gies Helmholtz Institute Jena & TPI, Friedrich-Schiller-Universität Jena & FRG @ Jena Functional Renormalization – from quantum gravity and dark energy to ultracold atoms and condensed matter Heidelberg, March 7-10 2017

FRG

ERG

E xact RG

E xact RG from first principles “includes irrelevant operators” but often only approximation solutions

NPRG

fun RG

ERG

E uropean RG

FRG

FRG: a prediction! (W ETTERICH ’93)

FRG: a prediction! (W ETTERICH ’93)

FRG: a prediction! (W ETTERICH ’93) . . . written in FRG Land

FRG: a prediction! (W ETTERICH ’93) . . . written in FRG Land . . . used only by IOC and FIFA

FRG: a prediction! (W ETTERICH ’93) . . . written in FRG Land . . . use discouraged by authorities . . . considered to be a derogatory communist term

From quantum gravity . . . to . . . condensed matter ✄ low dimensional relativistic fermions & quantum gravity ✄ (perturbative) QFT: � � δ ( γ ) = d − n E i [ φ i ] + n V α δ ( V α ) i α = ⇒ RG critical dimension: � 4 (gauge + matter, Yukawa/Higgs) D RG, cr = 2 (gravity, pure fermionic matter) ✄ many similarities: pert. nonrenormalizable, BUT: nonperturbatively renormalizable “Asymptotic safety” quantum phase transition

From quantum gravity . . . to . . . condensed matter ✄ low dimensional relativistic fermions & quantum gravity ✄ (perturbative) QFT: � � δ ( γ ) = d − n E i [ φ i ] + n V α δ ( V α ) i α = ⇒ RG critical dimension: � 4 (gauge + matter, Yukawa/Higgs) D RG, cr = 2 (gravity, pure fermionic matter) ✄ many similarities: pert. nonrenormalizable, BUT: nonperturbatively renormalizable “Asymptotic safety” quantum phase transition . . . no experimental evidence so far . . .

Chirality & Dirac Fermions ✄ d=3: { γ µ , γ ν } = 2 g µν , γ µ = 1 , 2 , 3 ∼ σ i = 1 , 2 , 3 (irreducible) = ⇒ no γ 5 (“no chirality”) ✄ Dirac fermions in irreducible representation: χ, ¯ χ 2-component

Chirality & Dirac Fermions ✄ d=3: (reducible, 4-comp. spinors: ψ, ¯ { γ µ , γ ν } = 2 g µν , ψ ) P L / R = 1 = ⇒ γ 5 , 2 ( 1 ± γ 5 ) & γ 4 & P 45 L / R = 1 γ 45 = i γ 4 γ 5 2 ( 1 ± γ 45 ) L kin = ¯ ψ a i ∂ /ψ a = ¯ ψ a /ψ a L + ¯ ψ a /ψ a L i ∂ R i ∂ R = . . . ✄ max. chiral symmetry group: U ( 2 N f ) chiral symmetry (reducible) ≃ flavor symmetry (irreducible)

Why 3 d chiral fermions? ✄ Goal: understanding QPTs with φ ↔ ψ, ¯ ψ order parameter gapless fermions . . . beyond the φ 4 paradigm (H ERBUT ’06) ✄ relativistic fermions from electrons on • honeycomb lattice • π -flux square lattice = ⇒ robust against weak interactions Hubbard model on honeycomb lattice (V OJTA ET AL .’00) Nodal d -wave superconductors (S ACHDEV ’10) ✄ for increasing coupling (Hubbard U or NN repulsion V ): phase transition: semi-metal → (Mott) insulator = ⇒ long-range order: AF, CDW, QAHS

Gross-Neveu model a.k.a. “chiral Ising” ✄ classical action, e.g., in d=3: a = 1 , . . . , N f � � � 1 / ψ a + ¯ g ( ¯ d 3 x ψ a i ∂ ψ a ψ a ) 2 S = ¯ , [¯ g ] = − 1 2 N f ✄ symmetries of reducible model: • discrete “chiral” symmetry: ψ a → γ 5 ψ a , ψ a → − ¯ ❩ 5 ¯ ψ a γ 5 2 : • flavor symmetry: L / R = 1 P 45 2 ( 1 ± γ 45 ) : U ( N f ) L × U ( N f ) R

Gross-Neveu model a.k.a. “chiral Ising” ✄ classical action, e.g., in d=3: a = 1 , . . . , 2 N f � � � 1 / χ a + d 3 x χ a i ∂ χ a χ a ) 2 S = ¯ g (¯ ¯ , [¯ g ] = − 1 2 N f ✄ symmetries of irreducible model: • parity symmetry: ❩ P χ a ( x ) → χ a ( − x ) , χ a ( x ) → − ¯ χ a ( − x ) 2 : ¯ • flavor symmetry: U ( 2 N f ) ✄ irreducible model in reducible notation (2 N f ∈ ◆ ): χ a χ a ) 2 ∼ ( ¯ ψγ 45 ψ ) 2 (¯

Gross-Neveu model a.k.a. “chiral Ising” ✄ classical action, e.g., in d=3: a = 1 , . . . , N f � � � 1 / ψ a + ¯ g ( ¯ d 3 x ψ a i ∂ ψ a ψ a ) 2 S = ¯ , [¯ g ] = − 1 2 N f ✄ discrete “chiral” symmetry: ψ a → γ 5 ψ a , ψ a → − ¯ ❩ 5 ¯ ψ a γ 5 2 :

Gross-Neveu model a.k.a. “chiral Ising” ✄ classical action, e.g., in d=3: a = 1 , . . . , N f � � � 1 / ψ a + ¯ g ( ¯ d 3 x ψ a i ∂ ψ a ψ a ) 2 S = ¯ , [¯ g ] = − 1 2 N f ✄ discrete “chiral” symmetry: ψ a → γ 5 ψ a , ψ a → − ¯ ❩ 5 ¯ ψ a γ 5 2 : ✄ Recette: On prend . . . (W ETTERICH ’93) ∂ t Γ k = 1 2 Tr ∂ t R k (Γ ( 2 ) + R k ) − 1 k

Gross-Neveu model Simplest approximation: “pointlike” vertices: � � � 1 / ψ a + ¯ g k ( ¯ d 3 x ψ a i ∂ ¯ ψ a ψ a ) 2 Γ k = 2 N f ✄ RG flow of dim’less coupling g = k d − 2 ¯ g k : ✄ UV fixed point: g ∗ ✄ IR divergence in scalar channel for g Λ > g ∗ indication for χ SB ✄ critical exponent Θ = 1 /ν = 1 (in d = 3) = ⇒ asymptotically safe proven to all orders in 1 / N f expansion (G AWEDZKI , K UPIAINEN ’85; R OSENSTEIN , W ARR , P ARK ’89; DE C ALAN ET AL .’91)

Partial Bosonization ✄ mapping to Yukawa model: (S TRATONOVICH ’58,H UBBARD ’59) � � 1 � / ψ a + ¯ g ( ¯ d 3 x ψ a i ∂ ψ a ψ a ) 2 ¯ S = 2 N f ↓ � � � h σ ) ψ a + N f m 2 σ 2 ¯ / + i ¯ d 3 x ψ a ( i ∂ S FB = 2 ¯ Pros: + RG flow into χ SB regime + access to long-range observables Cons: - use in FRG trunc’s: assumes dominance of bosonized channel - can be affected by “Fierz ambiguity” Cons less relevant for GN case

RG flow of Gross Neveu model (R OSA ,V ITALE ,W ETTERICH ’01; H OFLING ,N OWAK ,W ETTERICH ’02; B RAUN ,HG,S CHERER ’10) ✄ NLO derivative expansion: � � � h σ ) ψ a + 1 2 Z σ ( ∂ µ σ ) 2 + U ( σ ) Z ψ ¯ / + i ¯ ψ a ( i ∂ Γ k = ✄ quantum phase transition g Λ < g ∗ g Λ > g ∗

Exact large- N f fixed-point solution ✄ anomalous dimensions: (B RAUN ,HG,S CHERER ’10) η ψ = 0 , η σ = 1 ✄ large- N f fixed point effective potential for 2 < d < 4: , ρ = σ 2 u ∗ ( ρ ) = − 2 d − 8 � 1 − d 2 , 1 ; 2 − d 2 ; ( d − 4 )( d − 2 ) d � 3 d − 4 ρ 2 F 1 ρ 6 d − 8 d γ v d 2 ✄ exact critical exponents: Θ = 1 , − 1 , − 1 , − 3 , − 5 , − 7 , . . . = ⇒ critical surface: dim S = 1 physical parameter

Global effective potential and finite N f ✄ FP solver with pseudo-spectral methods (B ORCHARDT ,K NORR ’15)

3 d Gross-Neveu universality class, (arbitrary N f ) (B RAUN ,HG,S CHERER ’10) correlation exponent: ν = 1 Θ 1 ✄ leading-order derivative expansion identical results for irreducible model (R OSA ,V ITALE ,W ETTERICH ’01; H OFLING ,N OWAK ,W ETTERICH ’02)

FRG goes quantitative ✄ Derivative expansion: � � 1 ψψ + 1 2 Z ψ ( ρ )( ¯ ∂ψ − ( ∂ µ ¯ ψ ) γ µ ψ ) + h ( ρ ) ¯ 2 Z σ ( ρ )( ∂ µ σ ) 2 ψ/ Γ k = − U ( σ ) + iJ ψ ( ρ )( ∂ µ ρ ) ¯ ψγ µ ψ + X 1 ( ρ ) σ ( ∂ µ ¯ ψ )( ∂ µ ψ ) + i 2 X 2 ( ρ )( ∂ µ σ )[ ¯ ∂ψ − ( ∂ µ ¯ ψ ) γ µ ψ ] + X 3 ( ρ )( ∂ 2 σ ) ¯ ψ/ ψψ + 1 2 X 4 ( ρ )( ∂ µ σ )[ ¯ ψ Σ µν ∂ ν ψ − ( ∂ ν ¯ ψ )Σ µν ψ ] + 1 � 3 ( ρ )]( ∂ µ σ ) 2 σ ¯ 2 [ X 5 ( ρ ) + 2 X ′ ψψ • FRG LO: U ( ρ ) , h , Z ψ , Z σ (B RAUN ,HG,S CHERER ’10) • FRG LO’: U ( ρ ) , h ( ρ ) , Z ψ , Z σ (V ACCA ,Z AMBELLI ’15) • FRG NLO (K NORR ’16) (+regulator optimization, + pseudospectral solver + X A CT )

FRG goes quantitative (K NORR ’16)

FRG goes quantitative ✄ critical exponents N f = 2: FRG FRG FRG FRG FRG LO LO LO+ps LO’ NLO iGN rGN rGN rGN rGN (HNW’02) (BGS’10) (BK’15) (VZ’15) (K’16) ν 1.018 1.018 1.018 1.004 1.006(2) η σ 0.756 0.760 0.760 0.789 0.7765 η ψ 0.032 0.032 0.032 0.031 0.0276 (H OFLING ,N OWAK ,W ETTERICH ’02; B RAUN ,HG,S CHERER ’10; B ORCHARDT ,K NORR ’15; V ACCA ,Z AMBELLI ’15; K NORR ’16) = ⇒ satisfactory apparent convergence FRG performs rather well already at LO

FRG goes quantitative ✄ critical exponents N f = 2: method comparison FRG MC 1 / N f 2 + ǫ 2 + ǫ 4 − ǫ 2-sided Padé 3rd 4th +res. 2nd NLO (K’16) (KLLP’94) (G’94;HJ’14) (G’90’91;LR’91) (GLS’16) (RYK’93) (FGKT’16) ν 1.006(2) 1.00(4) 1.04 1.309 1.074 0.948 1.055 η σ 0.7765 0.754(8) 0.776 0.602 0.745 0.695 0.739 η ψ 0.0276 – 0.044 0.081 0.082 0.065 0.041 (K NORR ’16) (K ARKKAINEN ,L ACAZE ,L ACOCK ,P ETERSSON ’94) (G RACEY ’94; H ERBUT ,J ANSSEN ’14) (G RACEY ’90’91; L UPERINI ,R OSSI ’91) (G RACEY ,L UTH ,S CHRODER ’16) (R OSENSTEIN ,Y U ,K OVNER ’93) (F EI ,G IOMBI ,K LEBANOV ,T ARNOPOLSKY ’16) (P OSTER : B. I HRIG ) = ⇒ acceptable overall agreement with minor exceptions

FRG goes quantitative ✄ critical exponents N f = 1: method comparison