Minimally doubled chiral fermions Michael Creutz Brookhaven National Laboratory 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 Michael Creutz BNL 1
On the lattice ignoring the anomaly gives doublers • naive fermions: 16 species, exact U (4) L × U (4) R symmetry • staggered fermions: 4 species (tastes), one exact chiral symmetry • Wilson fermions: one light species • all chiral symmetries broken by doubler mass term • overlap, domain wall, perfect actions: N f arbitrary but • not ultra-local: computationally intensive • anomaly hidden, γ 5 � = ˆ γ 5 , Trˆ γ 5 = 2 ν � = 0 Michael Creutz BNL 2
Minimally doubled chiral fermion actions have just 2 species • Karsten 1981 • Wilczek 1987 • recent revival: MC, Borici, Bedaque Buchoff Tiburzi Walker-Loud Motivations • failure of rooting for staggered • lack of chiral symmetry for Wilson • computational demands of overlap, domain-wall approaches Elegant connection to the electronic structure of graphene • vanishing mass protected by topological considerations Michael Creutz BNL 3
Graphene: two dimensional hexagonal lattice of carbon atoms • http://online.kitp.ucsb.edu/online/bblunch/castroneto/ • A. H. Castro Neto et al., arXiv:0709.1163 Held together by strong ‘‘sigma’’ bonds, sp 2 One ‘‘pi’’ electron per site can hop around Consider only nearest neighbor hopping in the pi system • tight binding approximation Michael Creutz BNL 4
Fortuitous choice of coordinates helps solve x 2 x 1 a b Form horizontal bonds into ‘‘sites’’ involving two types of atom • ‘‘ a ’’ on the left end of a horizontal bond • ‘‘ b ’’ on the right end • all hoppings are between type a and type b atoms Label sites by non-orthogonal coordinates x 1 and x 2 • axes at 30 degrees from horizontal Michael Creutz BNL 5
Hamiltonian a b � a † x 1 ,x 2 b x 1 ,x 2 + b † H = K x 1 ,x 2 a x 1 ,x 2 a x 1 ,x 2 b + a † x 1 +1 ,x 2 b x 1 ,x 2 + b † x 1 − 1 ,x 2 a x 1 ,x 2 b + a † x 1 ,x 2 − 1 b x 1 ,x 2 + b † x 1 ,x 2 +1 a x 1 ,x 2 a • hops always between a and b sites Go to momentum (reciprocal) space � π 2 π e ip 1 x 1 e ip 2 x 2 ˜ dp 1 dp 2 • a x 1 ,x 2 = a p 1 ,p 2 . − π 2 π • − π < p µ ≤ π Michael Creutz BNL 6
Hamiltonian breaks into two by two blocks � � ˜ � π dp 1 dp 2 � � a p 1 ,p 2 0 z ˜ a † b † H = K ( ˜ p 1 ,p 2 ) ˜ p 1 ,p 2 z ∗ 0 b p 1 ,p 2 2 π 2 π − π • where z = 1 + e − ip 1 + e + ip 2 a b a b b a � � 0 z ˜ H ( p 1 , p 2 ) = K z ∗ 0 Fermion energy levels at E ( p 1 , p 2 ) = ± K | z | • energy vanishes only when | z | does • exactly two points p 1 = p 2 = ± 2 π/ 3 Michael Creutz BNL 7
Topological stability • contour of constant energy near a zero point • phase of z wraps around unit circle • cannot collapse contour without going to | z | = 0 p2 π 2π/3 E E p1 −π 2π/3 −2π/3 π p p −2π/3 −π allowed forbidden No band gap allowed • Graphite is black and a conductor Michael Creutz BNL 8
No-go theorem Nielsen and Ninomiya • periodicity of Brillouin zone • wrapping around one zero must unwrap elsewhere • two zeros is the minimum possible Connection with chiral symmetry • b → − b changes sign of H � � � � 0 z 1 0 anticommutes with σ 3 = ˜ • H ( p 1 , p 2 ) = K z ∗ 0 0 − 1 • σ 3 → γ 5 in four dimensions Michael Creutz BNL 9
Four dimensions Want Dirac operator D to put into path integral action ψDψ • require ‘‘ γ 5 Hermiticity’’ • γ 5 Dγ 5 = D † • work with Hermitean ‘‘Hamiltonian’’ H = γ 5 D • not the Hamiltonian of the 3D Minkowski theory Require same form as the two dimensional case � � 0 z ˜ H ( p µ ) = K z ∗ 0 • four component momentum, ( p 1 , p 2 , p 3 , p 4 ) Michael Creutz BNL 10
a To keep topological argument 0 • extend z to quaternions • z = a 0 + i� a · � σ a • | z | 2 = � µ a 2 µ H ( p µ ) now a four by four matrix ˜ • ‘‘energy’’ eigenvalues still E ( p µ ) = ± K | z | • constant energy surface topologically an S 3 • surrounding a zero should give non-trivial mapping Michael Creutz BNL 11
Implementation • not unique • here I follow Borici’s construction 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 • Dirac operator D anticommutes with γ 5 • an exact chiral symmetry • part of an exact SU (4) × SU (4) chiral algebra Karsten and Smit Michael Creutz BNL 12
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’’ • ‘‘doublers’’ Michael Creutz BNL 13
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 Michael Creutz BNL 14
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 Michael Creutz BNL 15
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! Michael Creutz BNL 16
(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 ) • at other zeros of D , D − 4 i Γ is large • at other zeros of D , D − 4 i Γ is large Michael Creutz BNL 17
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 Michael Creutz BNL 18
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 Michael Creutz BNL 19
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 Michael Creutz BNL 20
Natural time axis along main diagonal e 1 + e 2 + e 3 + e 4 • T exchanges the two Fermi points • increases symmetry group to 48 elements Karsten and Wilczek actions • e 4 as the special direction Charge conjugation: equivalent to particle hole symmetry • D and H = γ 5 D have eigenvalues in opposite sign pairs Michael Creutz BNL 21
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 Michael Creutz BNL 22
Issues and questions Requires a multiple of two flavors • can split degeneracies with Wilson terms Only one exact chiral symmetry • not the full SU (2) ⊗ SU (2) • enough to protect mass • π 0 a Goldstone boson • π ± only approximate Not unique • only need z ( p ) with two zeros • above: Borici’s variation with orthogonal coordinates • alternatives: Karsten, Wilczek, MC Michael Creutz BNL 23
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