Graphene and chiral fermions Michael Creutz BNL & U. Mainz - PowerPoint PPT Presentation

Graphene and chiral fermions Michael Creutz BNL & U. Mainz Extending graphene structure to four dimensions gives • a two-flavor lattice fermion action • one exact chiral symmetry • protects mass renormalization • strictly local action • only nearest neighbor hopping • fast for simulations Michael Creutz BNL & U. Mainz 1

Graphene electronic structure remarkable low excitations described by a massless Dirac equation • two ‘‘flavors’’ of excitation • versus four of naive lattice fermions • • massless structure robust • relies on a ‘‘chiral’’ symmetry • involves mapping circles onto circles Four dimensional extension 3 coordinate carbon replaced by 5 coordinate ‘‘atoms’’ • generalize topology to mapping spheres onto spheres • complex numbers replaced by quaternions • Michael Creutz BNL & U. Mainz 2

Chiral symmetry versus the lattice • Lattice is a regulator • removes all infinities • continuum limit defines a field theory Classical U (1) chiral symmetry broken by quantum effects • a valid lattice formulation must break U (1) axial symmetry • But we want flavored chiral symmetries to protect masses • Wilson fermions break all these • staggered require four flavors for one chiral symmetry • overlap, domain wall non-local, computationally intensive • Graphene fermions do it in the minimum way allowed! Michael Creutz BNL & U. Mainz 3

The graphene structure A two dimensional hexagonal planar structure of carbon atoms A. H. Castro Neto et al., RMP 81,109 [arXiv:0709.1163] • http://online.kitp.ucsb.edu/online/bblunch/castroneto/ • 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 & U. Mainz 4

Fortuitous choice of coordinates helps solve x x 2 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’’ with non-orthogonal coordinates x 1 and x 2 axes at 30 degrees from horizontal • Michael Creutz BNL & U. Mainz 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 . 2 π − π • − π < p µ ≤ π Michael Creutz BNL & U. Mainz 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 ) ˜ z ∗ p 1 ,p 2 0 b p 1 ,p 2 2 π 2 π − π + e + ip 2 z = 1 + e − ip 1 where • 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 when | z | does • exactly two points • p 1 = p 2 = ± 2 π/ 3 Michael Creutz BNL & U. Mainz 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 • p 2 π 2π/3 E E p 1 −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 & U. Mainz 8

Connection with chiral symmetry • b → − b changes sign of H � � 0 z ˜ • H ( p 1 , p 2 ) = K z ∗ 0 � � 1 0 anticommutes with σ 3 = • 0 − 1 • σ 3 → γ 5 in four dimensions No-go theorem Nielsen and Ninomiya (1981) periodicity of Brillouin zone • wrapping around one zero must unwrap elsewhere • two zeros is the minimum possible • Michael Creutz BNL & U. Mainz 9

Four dimensions Feynman path integral in temporal box of length T ( dA dψ dψ ) e − S = Tr e − Ht � • Z = � 1 ‘‘action’’ S = d 4 x � � • 4 F µν F µν + ψDψ Wick rotation to imaginary time: e iHT → e − HT • four coordinates x, y, z, t • Need Dirac operator D to put into path integral action ψDψ properties: D † = − D = γ 5 Dγ 5 ‘‘ γ 5 Hermiticity’’ • work with Hermitean ‘‘Hamiltonian’’ H = γ 5 D • not the Hamiltonian of the 3D Minkowski theory • Michael Creutz BNL & U. Mainz 10

Look for analogous form to the two dimensional case � � 0 z ˜ H ( p µ ) = K z ∗ 0 • z ( p 1 , p 2 , p 3 , p 4 ) depends on the four momentum components To keep topological argument a 0 • extend z to quaternions • z = a 0 + i� a · � σ a • | z | 2 = � µ a 2 µ Michael Creutz BNL & U. Mainz 11

˜ 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 • Introduce gamma matrix convention [ γ µ , γ ν ] + = 2 δ µν � � 0 � σ � γ = σ x ⊗ � σ = � σ 0 • � � 0 i γ 4 = − σ y ⊗ 1 = − i 0 � � 1 0 γ 5 = σ z ⊗ 1 = γ 1 γ 2 γ 3 γ 4 = 0 − 1 Michael Creutz BNL & U. Mainz 12

Continuum Dirac action D = ik µ γ µ � � 0 z γ 5 D = H = z ∗ 0 z = k 0 + i� k · � σ Lattice implementation • not unique • local action • only sines and cosines mimic 2-d case • 1 + e − ip 1 + e ip 2 = 1 + cos( p 1 ) + cos( p 2 ) − i (sin( p 1 ) − sin( p 2 )) Michael Creutz BNL & U. Mainz 13

Try z = B (4 C − cos( p 1 ) − cos( p 2 ) − cos( p 3 ) − cos( p 4 )) + iσ x (sin( p 1 ) + sin( p 2 ) − sin( p 3 ) − sin( p 4 )) + iσ y (sin( p 1 ) − sin( p 2 ) − sin( p 3 ) + sin( p 4 )) + iσ z (sin( p 1 ) − sin( p 2 ) + sin( p 3 ) − sin( p 4 )) B and C are constants to be determined control anisotropic distortions • similar to non-orthogonal coordinates in graphene solution • Michael Creutz BNL & U. Mainz 14

Zero of z requires all components to vanish, four relations sin( p 1 ) + sin( p 2 ) − sin( p 3 ) − sin( p 4 ) = 0 sin( p 1 ) − sin( p 2 ) − sin( p 3 ) + sin( p 4 ) = 0 sin( p 1 ) − sin( p 2 ) + sin( p 3 ) − sin( p 4 ) = 0 cos( p 1 ) + cos( p 2 ) + cos( p 3 ) + cos( p 4 ) = 4 C first three imply sin( p i ) = sin( p j ) ∀ i, j • • cos( p i ) = ± cos( p j ) last relation requires C < 1 • if C > 1 / 2 , only two solutions • • p i = p j = ± arccos( C ) Michael Creutz BNL & U. Mainz 15

As in two dimensions • expand about zeros • identify Dirac spectrum • rescale for physical momenta Expanding about the positive solution • p µ = ˜ p + q µ • p = arccos( C ) ˜ Michael Creutz BNL & U. Mainz 16

Reproduces the Dirac equation D = iγ µ k µ if we take k 1 = C ( q 1 + q 2 − q 3 − q 4 ) k 2 = C ( q 1 − q 2 − q 3 + q 4 ) k 3 = C ( q 1 − q 2 + q 3 − q 4 ) k 4 = BS ( q 1 + q 2 + q 3 + q 4 ) √ here S = sin(˜ 1 − C 2 • p ) = Other zero at ˜ p = − arccos( C ) flips sign of γ 4 • the two species have opposite chirality • the exact chiral symmetry is a flavored one • Michael Creutz BNL & U. Mainz 17

B and C control distortions between the k and q coordinates The k coordinates should be orthogonal • the q ’s are not in general • = B 2 S 2 − C 2 q i · q j B 2 S 2 + 3 C 2 | q | 2 If B = C/S the q axes are also orthogonal allows gauging with simple plaquette action • √ Borici: B = 1 , C = S = 1 / • 2 Michael Creutz BNL & U. Mainz 18

Alternative choice for B and C from graphene analogy zeros of z in periodic momentum space form a lattice • give each zero 5 symmetrically arranged neighbors • √ • C = cos( π/ 5) , B = 5 interbond angle θ satisfies cos( θ ) = − 1 / 4 • • θ = acos( − 1 / 4) = 104 . 4775 . . . degrees 4-d generalization of the diamond lattice • Michael Creutz BNL & U. Mainz 19

The physical lattice structure Graphene: one bond splits into two in two dimensions • θ = acos( − 1 / 2) = 120 degrees iterating smallest loops are hexagons • Michael Creutz BNL & U. Mainz 20

Diamond: one bond splits into three in three dimensions tetrahedral environment • • θ = acos( − 1 / 3) = 109 . 4712 . . . degrees iterating smallest loops are cyclohexane chairs • Michael Creutz BNL & U. Mainz 21

4-d graphene ‘‘hyperdiamond’’: one bond splits into four 5-fold symmetric environment • • θ = acos( − 1 / 4) = 104 . 4775 . . . degrees iterating smallest loops are hexagonal ‘‘chairs’’ • Michael Creutz BNL & U. Mainz 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 from additive renormalization • only one Goldstone boson: π 0 • • π ± only approximate One direction treated differently Bedaque, Buchoff, Tibursi, Walker-Loud • γ 4 has a different phase from the spatial gammas with interactions lattice can distort along one direction • requires tuning anisotropy • Michael Creutz BNL & U. Mainz 23