Graphene and chiral fermions

Transcription

1 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

2 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

3 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

4 The graphene structure A two dimensional hexagonal planar structure of carbon atoms A. H. Castro Neto et al., RMP 81,109 [arxiv: ] 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

5 Fortuitous choice of coordinates helps solve x 2 1 x 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

6 Hamiltonian H = K a x 1,x 2 b x1,x 2 + b x 1,x 2 a x1,x 2 x 1,x 2 +a x 1 +1,x 2 b x1,x 2 + b x 1 1,x 2 a x1,x 2 +a x 1,x 2 1 b x 1,x 2 + b x 1,x 2 +1 a x 1,x 2 hops always between a and b sites a b b a a b Go to momentum (reciprocal) space a x1,x 2 = π π π < p µ π dp 1 2π dp 2 2π eip 1x 1 e ip 2x2 ã p1,p 2. Michael Creutz BNL & U. Mainz 6

7 Hamiltonian breaks into two by two blocks H = K π π dp 1 2π dp 2 2π ( ã p 1,p 2 b p1,p 2 ) where z = 1 + e ip 1 + e +ip 2 a b a ( ) ( ) 0 z ãp1,p 2 z 0 bp1,p 2 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 = ±2π/3 Michael Creutz BNL & U. Mainz 7

8 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 π 2π/3 2π/3 π p 1 E p E p 2π/3 π allowed forbidden No band gap allowed Graphite is black and a conductor Michael Creutz BNL & U. Mainz 8

9 Connection with chiral symmetry b b changes sign of H ( ) H(p 0 z 1,p 2 ) = K z 0 anticommutes with σ 3 = 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

10 Four dimensions Feynman path integral in temporal box of length T Z = (da dψ dψ)e S = Tr e Ht action S = d 4 x ( 1 4 F µνf 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

11 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 extend z to quaternions z = a 0 + i a σ z 2 = µ a2 µ a 0 a Michael Creutz BNL & U. Mainz 11

12 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 ] + = 2δ 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

13 Continuum Dirac action D = ik µ γ µ γ 5 D = H = z = k 0 + i k σ ( 0 ) z z 0 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

14 Try z =B(4C 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

15 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 ) = 4C 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

16 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

17 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( p) = 1 C 2 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

18 B and C control distortions between the k and q coordinates The k coordinates should be orthogonal the q s are not in general q i q j q 2 = B2 S 2 C 2 B 2 S 2 + 3C 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

19 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) = degrees 4-d generalization of the diamond lattice Michael Creutz BNL & U. Mainz 19

20 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

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

22 4-d graphene hyperdiamond : one bond splits into four 5-fold symmetric environment θ = acos( 1/4) = degrees iterating smallest loops are hexagonal chairs Michael Creutz BNL & U. Mainz 22

23 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

24 Not unique only need z(p) with two zeros Here C = cos(π/5), B = 5 gives approximate 120 element pentahedral symmetry Borici s variation with orthogonal coordinates a linear combination of two naive fermion formulations Michael Creutz BNL & U. Mainz 24

25 Karsten (1981) and Wilczek (1987) select the time axis as special like spatial Wilson fermions with r irγ 0 Karsten and Wilczek forms equivalent up to phases Tatsuhiro Misumi D =iγ 1 (sin(p 1 ) + cos(p 2 ) 1) iγ 2 (sin(p 2 ) + cos(p 3 ) 1) iγ 3 (sin(p 3 ) + cos(p 4 ) 1) iγ 4 (sin(p 4 ) + cos(p 1 ) 1) poles at p = (0,0,0,0) and p = (π/2,π/2,π/2,π/2) Michael Creutz BNL & U. Mainz 25

26 Gauge field topology and zero modes the two flavors have opposite chirality their respective zero modes can mix through lattice artifacts no longer exact zero eigenvalues of D similar to staggered, but 2 rather than 4 flavors Comparison with staggered both have one exact chiral symmetry both have only approximate zero modes from topology four component versus one component fermion field two versus four flavors (tastes) no uncontrolled extrapolation to two physical light flavors Michael Creutz BNL & U. Mainz 26

27 Perturbative corrections can shift pole positions Capitani, Weber, Wittig shift along direction between the poles Generalized Karsten/Wilczek operator: D = iγ 4 sin(α) ( 4 poles at p = 0, p 4 = ±α ) µ=1 cos(p µ) cos(α) 3 +i 3 i=1 γ i sin(p i ) alpha gets an additive renormalization tune coefficient of ψγ 4 ψ dimension 3 Two operators control asymmetry ψγ 4 4 ψ and β t dimension 4 Michael Creutz BNL & U. Mainz 27

28 Point split fields natural separate poles at different bare momenta u(q) = 1 ( 1 + sin(q ) 4 + α) ψ(q + αe 4 ) 2 sin(α) d(q) = 1 ( 2 Γ 1 sin(q ) 4 α) ψ(q αe 4 ) sin(α) zeros inserted to cancel undesired pole not unique Γ factor since different poles use different gamma matrices Γ = iγ 4 γ 5 for Karsten/Wilczek formulation Michael Creutz BNL & U. Mainz 28

29 Position space: u(x) = 1 ( 2 eiαx 4 ψ(x) + i ψ(x e ) 4) ψ(x + e 4 ) 2 sin(α) d(x) = 1 ( 2 Γe iαx 4 ψ(x) i ψ(x e ) 4) ψ(x + e 4 ) 2 sin(α) Gives rise to point-split meson operators; i.e. η (x) = 1 ( ψ(x e 4 )γ 5 ψ(x) ψ(x)γ 5 ψ(x e 4 ) 8 ) + ψ(x + e 4 )γ 5 ψ(x) ψ(x)γ 5 ψ(x + e 4 ). Michael Creutz BNL & U. Mainz 29

30 Effective Lagrangians and lattice artifacts MC, Sharpe and Singleton Two possibilities for Wilson fermions as m q 0 Chiral transition becomes first order Aoki phase Two choices here as well m π± > m π0 : π 0 is normal Goldstone mode m π± < m π0 : 2nd order transition before m q 0 paired eigenvalues imply a positive fermion determinant Vafa-Witten argument suggests first option Michael Creutz BNL & U. Mainz 30

31 Summary Extending graphene and diamond lattices to four dimensions: a two-flavor lattice Dirac operator one exact chiral symmetry protects from additive mass renormalization eigenvalues purely imaginary for massless theory in complex conjugate pairs strictly local fast to simulate Michael Creutz BNL & U. Mainz 31

32 Extra Slides Michael Creutz BNL & U. Mainz 32

33 Valence bond theory for carbon Carbon has 6 electrons two tightly bound in the 1s orbital second shell: one 2s and three 2p orbitals In a molecule or crystal, external fields mix the 2s and 2p orbitals Carbon likes to mix the outer orbitals in two distinct ways 4 sp 3 orbitals in a tetrahedral arrangement methane CH 4, diamond C 3 sp 2 orbitals in a planar triangle plus one p benzene C 6 H 6, graphite C the sp 2 electrons in strong sigma bonds the p electron can hop around in pi orbitals H H H C C C C H H C C C H H H H H Michael Creutz BNL & U. Mainz 33

34 Hexagonal structure hidden in deformed coordinates p 2 π π π p 1 π Thomas Szkopek Michael Creutz BNL & U. Mainz 34

35 Position space rules from identifying e ±ip terms with hopping on site action: 4iBCψγ 4 ψ hop in direction 1: ψ j (+γ 1 + γ 2 + γ 3 ibγ 4 )ψ i hop in direction 2: ψ j (+γ 1 γ 2 γ 3 ibγ 4 )ψ i hop in direction 3: ψ j ( γ 1 γ 2 + γ 3 ibγ 4 )ψ i hop in direction 4: ψ j ( γ 1 + γ 2 γ 3 ibγ 4 )ψ i minus the conjugate for a reverse hop Notes a mixture real and imaginary coefficients for the γ s γ 5 exactly anticommutes with D D is purely anti-hermitean γ 4 not symmetrically treated to γ Michael Creutz BNL & U. Mainz 35