Local chiral fermions Michael Creutz Brookhaven National Laboratory - PowerPoint PPT Presentation

Local chiral fermions Michael Creutz Brookhaven National Laboratory Summary • A strictly local fermion action D ( A ) • with one exact chiral symmetry γ 5 D = −D γ 5 • describing two flavors; minimum required for chiral symmetry • a linear combination of two ‘‘naive’’ fermion actions (Borici) • Space-time symmetries • translations plus 48 element subgroup of hypercubic rotations • includes odd parity transformations • renormalization can induce anisotropy at finite a 1 Michael Creutz BNL

Chiral symmetry crucial to our understanding of hadronic physics • pions are waves on a background quark condensate � ψψ � • chiral extrapolations essential to practical lattice calculations Anomaly removes classical U (1) chiral symmetry • SU ( N f ) × SU ( N f ) × U B (1) • non trivial symmetry requires N f ≥ 2 Minimally doubled chiral fermion actions have just 2 species • Karsten 1981 • Wilczek 1987 • recent revival: MC, Borici, Bedaque Buchoff Tiburzi Walker-Loud 2 Michael Creutz BNL

Motivations • failure of rooting for staggered • lack of chiral symmetry for Wilson • computational demands of overlap, domain-wall approaches Here I follow Borici’s construction • linear combination of two equivalent naive fermion actions 3 Michael Creutz BNL

Start with naive fermions • forward hop between sites γ µ U unit hopping parameter for convenience • backward hop between sites − γ µ U † • µ is the direction of the hop • U is the usual gauge field matrix • 16 doublers • Dirac operator D anticommutes with γ 5 • an exact chiral symmetry • part of an exact SU (4) × SU (4) chiral algebra Karsten and Smit 4 Michael Creutz BNL

In the free limit, solution in momentum space � D ( p ) = 2 i γ µ sin( p µ ) µ • for small momenta reduces to Dirac equation • 15 extra Dirac equations for components of momenta near 0 or π (π,π) (π,π) (π,0) (π,0) X p p y t (0,0) (0,π) (0,0) (0,π) p x p z 16 ‘‘Fermi points’’ 5 Michael Creutz BNL

Consider momenta maximally distant from the zeros: p µ = ± π/ 2 (π,π) (π,0) (−π/2,π/2) (π/2,π/2) (−π/2,−π/2) p y (π/2,−π/2) (0,0) (0,π) p x Select one of these points, i.e. p µ = + π/ 2 for every µ • D ( p µ = π/ 2) = 2 i � µ γ µ ≡ 4 i Γ • Γ ≡ 1 2 ( γ 1 + γ 2 + γ 3 + γ 4 ) • unitary, Hermitean, traceless 4 by 4 matrix 6 Michael Creutz BNL

Now consider a unitary transformation • ψ ′ ( x ) = e − iπ ( x 1 + x 2 + x 3 + x 4 ) / 2 Γ ψ ( x ) ′ ( x ) = e iπ ( x 1 + x 2 + x 3 + x 4 ) / 2 ψ ( x ) Γ • ψ • phases move Fermi points from p µ ∈ { 0 , π } to p µ ∈ {± π/ 2 } • ψ ′ uses new gamma matrices γ ′ µ = Γ γ µ Γ • Γ = 1 2 ( γ 1 + γ 2 + γ 3 + γ 4 ) = Γ ′ • new free action: µ γ ′ D ( p ) = 2 i � µ sin( π/ 2 − p µ ) D and D physically equivalent 7 Michael Creutz BNL

Complimentarity: D ( p µ = π/ 2) = D ( p µ = 0) = 4 i Γ Combine the naive actions D = D + D − 4 i Γ Free theory � γ µ sin( p µ ) + γ ′ � • D ( p ) = 2 i � µ sin( π/ 2 − p µ ) − 4 i Γ µ • at p µ ∼ 0 the 4 i Γ term cancels D , leaving D ( p ) ∼ γ µ p µ • at p µ ∼ π/ 2 the 4 i Γ term cancels D , leaving D ( π/ 2 − p ) ∼ γ ′ µ p µ • Only these two zeros of D ( p ) remain! 8 Michael Creutz BNL

(π,π) (π,0) (−π/2,π/2) (π/2,π/2) (−π/2,−π/2) p y (π/2,−π/2) (0,0) (0,π) p x THEOREM: these are the only zeros of D ( p ) (appendix) • at other zeros of D , D − 4 i Γ is large • at other zeros of D , D − 4 i Γ is large 9 Michael Creutz BNL

Chiral symmetry remains exact • γ 5 D = −D γ 5 • e iθγ 5 D e iθγ 5 = D But • γ ′ 5 = Γ γ 5 Γ = − γ 5 • two species rotate oppositely • symmetry is flavor non-singlet 10 Michael Creutz BNL

Space time symmetries • usual discrete translation symmetry µ γ µ treats primary hypercube diagonal specially • Γ = 1 � 2 • action symmetric under subgroup of the hypercubic group • leaving this diagonal invariant • includes Z 3 rotations amongst any three positive directions √ • V = exp(( iπ/ 3)( σ 12 + σ 23 + σ 31 ) / 3) [ γ µ , γ ν ] + = 2 iσ µν • cyclicly permutes x 1 , x 2 , x 3 axes [ V, Γ] = 0 • physical rotation by 2 π/ 3 z y x y z x • V 3 = − 1 : we are dealing with fermions 11 Michael Creutz BNL

Repeating with other axes generates the 12 element tetrahedral group • subgroup of the full hypercubic group Odd-parity transformations double the symmetry group to 24 elements 1 • V = 2 (1 + iσ 15 )(1 + iσ 21 )(1 + iσ 52 ) [ V, Γ] = 0 √ 2 • permutes x 1 , x 2 axes • γ 5 → V † γ 5 V = − γ 5 z z y x y x 12 Michael Creutz BNL

Natural time axis along main diagonal e 1 + e 2 + e 3 + e 4 • T exchanges the Fermi points • increases symmetry group to 48 elements Charge conjugation: equivalent to particle hole symmetry • D and H = γ 5 D have eigenvalues in opposite sign pairs 13 Michael Creutz BNL

Special treatment of main diagonal • interactions can induce lattice distortions along this direction 1 • a (cos( ap ) − 1) ψ Γ ψ = O ( a ) • symmetry restored in continuum limit • at finite lattice spacing can tune Bedaque Buchoff Tiburzi Walker-Loud • coefficient of iψ Γ ψ dimension 3 operator • 6 link plaquettes orthogonal to this diagonal • zeros topologically robust under such distortions • Nielsen Ninomiya, MC 14 Michael Creutz BNL

Appendix A: Proof that there are only two zeros of D ( p ) • Tr ( γ µ − γ ν ) D ( p ) ∼ sin( p µ − π/ 4) − sin( p ν − π/ 4) • at a zero: cos( p µ − π/ 4) = ± cos( p ν − π/ 4) • all cosines equal in magnitude √ • Tr Γ D ( p ) = 0 ⇒ � µ cos( p µ − π/ 4) = 2 2 > 2 • all cosines positive √ • at a zero: cos( p µ − π/ 4) = +1 / 2 All components of p µ are equal and either 0 or π/ 2 15 Michael Creutz BNL

Appendix B: Actions from Karsten and Wilczek • both equivalent up to a unitary transformation • ψ → i x 4 ψ 4 3 � � D = γ µ sin( p µ ) + γ 4 (1 − cos( p i )) µ =1 i =1 • last term removes all zeros except � p = 0 , p 4 = 0 , π Now x 4 chosen as the special direction • onsite term ∼ γ 4 instead of ∼ Γ γ ′ = � γ , γ ′ • � 4 = − γ 4 • γ ′ 5 = − γ 5 16 Michael Creutz BNL

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