Emergent spin Michael Creutz BNL Introduction quantum mechanics + relativity spin statistics connection fermions have half integer spin On a lattice no relativity can consider spinless fermions hopping around If low energy excitations have a relativistic spectrum spin must emerge from the dynamics Michael Creutz BNL 1
Lattice examples graphene D = 2 (Mecklenburg and Regen, PRL) staggered fermionsd = 2,3,4 In the continuum fermions can appear as topological objects bosonization Skyrmions Michael Creutz BNL 2
Graphene 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 3
Fortuitous choice of coordinates helps solve Hopping on a hexagonal lattice 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 Label sites with non-orthogonal coordinatesx 1 andx 2 axes at 30 degrees from horizontal Michael Creutz BNL 4
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,x 2 a x1 +1,x 2 +a x 1,x 2 b x1,x 2 +1 +b x 1,x 2 +1 a x 1,x 2 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 5
Hamiltonian breaks into two by two blocks H = K π π dp 1 2π ( ) dp 2 0 z )(ãp1 2π (ã,p p 1,p 2 b p1,p 2 ) 2 z 0 bp1,p 2 where z = 1 +e ip 1 +e +ip 2 a b a b b a Spinorψ = ( a b ) emerges Michael Creutz BNL 6
H(p 1,p 2 ) = K ( ) 0 z z 0 z = 1 +e ip 1 +e +ip 2 eigenstates are two component spinors ( ψ = z 1 2 z ± z E(p 1,p 2 ) = ±K z ) energy vanishes when z does exactly two points p 1 = p 2 = ±2π/3 Michael Creutz BNL 7
Topology and spin contour of constant energy near a zero point phase ofz 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 ψ ψ on2π rotation of momentum around a zero half integer spin emerges! Michael Creutz BNL 8
two dimensional lattice Hopping in a magnetic field H = K {i,j} a i Z ija j +a j Z ij a i Z = e iφ U(1) 2 3 constantb = Re P Z 1 4 Michael Creutz BNL 9
General B gives a fractal structure Hofstadter s butterfly B = p/q givesq bands Douglas Hofstadter Concentrate onb = 1/2 every plaquette givesz P = 1 two bands, touching at two Dirac cones Michael Creutz BNL 10
Choose a convenient gauge Z x = 1,Z y = ( 1) x two types of site a b a b a 1 1 1 1 1 1 1 1 1 a b 1 1 1 1 1 1 1 1 a b 1 1 1 1 1 1 1 1 a b 1 1 1 1 1 1 1 1 1 1 a b 1 1 1 1 1 a on top of negativey bonds b on top of positivey bonds Michael Creutz BNL 11
Periodicity by 2 in x direction by 1 in y direction Momentum space ψ(2n,m) = e 2ipn+iqm ψ(0,0) ψ(2n+1,m) = e 2ipn+iqm ψ(1,0) ( ) ψ(0,0) two component base spinor Ψ = ψ(1, 0) 0 q < 2π 0 p < π Half sized Brilloin zone Michael Creutz BNL 12
A two by two Hamiltonian matrix ( 2cos(q) 1+e 2ip H = K 1+e 2ip 2cos(q) ) E = ±2K cos 2 (p)+cos 2 (q) zeros at(p,q) = (π/2,π/2) and(p,q) = (π/2,3π/2) two Dirac cones as with graphene This is a rewriting of staggered fermions Michael Creutz BNL 13
Three dimensions Thread half integer magnetic flux through every plaquette convenient gauge gives 4 types of site Z x = 1, Z y = ( 1) x, Z z = ( 1) x+y translate to a four component base spinor ψ(0,0,0) ψ(1,0,0) ψ(0,1,0) ψ(1,1,0) E = 2K cos 2 (p x )+cos 2 (p y )+cos 2 (p z ) p x andp y (notp z ) restricted to half zones0 p < π Equivalent to 3D staggered Hamiltonian Michael Creutz BNL 14
Chiral symmetry Hamiltonian anticommutes with( 1) x+y+z γ 5 two dirac cones have opposite chirality a two flavor theory with one exact chiral symmetry actually a flavored chiral symmetry consistent with anomaly Michael Creutz BNL 15
Adding non-abelian gauge fields Spinless fermion coupled to two gauge fields SU(N) of color;u ij on links auxiliaryz 2 ; Z ij on links fermion hop picks up productu ij Z ij Two gauge couplings β 1 g 2 for the color group β z for thez 2 Take the limitβ z forces eachz 2 plaquette to 1 This is staggered Hamiltonian lattice gauge theory Periodicity of Brilloin zone implies two cones fermion doubling -- chiral symmetry is flavored Michael Creutz BNL 16
SU(2) color 1 SU(2) can absorb phases in the group links flipping sign of the gauge couplingβ β SU(2) Hamiltonian lattice gauge theory of spinless fermions at negative β is equivalent to staggered fermions! Not true forsu(3) since 1 not in the group Michael Creutz BNL 17
Four dimensions Staggered formulated as above brings in one extra doubling effective 8 component spinor four tastes 3d Hamiltonian has only two tastes Can we insert the 3D Hamiltonian in a 4D path integral to get a two taste formulation of staggered? Michael Creutz BNL 18
Fermionic path integrals connect Hamiltonian and Euclidian path integral formulations consider a product of normal ordered operators Tr ( : f 1 (a,a) :: f 2 (a,a) :... : f n (a,a) : ) = (dψdψ )f 1 (ψ1,ψ 1)...f n (ψn,ψ n)e j ψ j (ψ j ψ j 1 ) where ψ n andψn ψ 0 = ψ n are independent Grassmann variables antiperiodic boundaries note the asymmetric discrete derivative Michael Creutz BNL 19
Applied totr ( : e βh/n t : ) N t using above 3D Hamiltonian gives a Fermion action minimally doubled with two tastes minimal number required for chiral symmetry But: different time treatment breaks 4d hypercubic symmetry contains both Hermitean and anti-hermitean parts All known 4d minimally doubled chiral formulations break hypercubic symmetry is there a theorem? Michael Creutz BNL 20
Gauge fields and topology Index theorem:ν = n + n n ± zero modes of Dirac operator of±chirality ν topological index of the gauge field formallytr γ 5 = ν Singlet chiral symmetry is anomalous ψ e iγ 5θ ψ changes integration measuredψ e iνθ dψ m ψψ mψe iγ 5θ ψ an inequivalent theory; theta vacuum violates CP symmetry Michael Creutz BNL 21
Consider any lattice Dirac operator D assume gamma five hermiticityγ 5 Dγ 5 = D all operators in practice satisfy this (except twisted mass) Divide D into hermitean and antihermitean parts D = K +M K = (D D )/2 M = (D +D )/2 Michael Creutz BNL 22
Then [K,γ 5 ] + = 0 [M,γ 5 ] = 0 M e iθγ 5 M an exact symmetry of the determinant Where is the anomaly? Earlier constructions solve this with doublers half useγ 5 and half γ 5 the naive chiral symmetry is actually flavored Michael Creutz BNL 23
How about Wilson fermions? doublers given masses of order the cutoff the rotationm e iθγ 5 M also rotates their phases PhysicalΘis a relative angle independently rotate the fermion mass and the Wilson term Seiler and Stamatescu The overlap operator eigenvalues on a circle zero eigenmodes have heavy counterpart rotation of Hermitean part rotates heavy mode as well anomaly brings in ˆγ 5 : Dγ 5 = ˆγ 5 D ν = Tr(γ 5 + ˆγ 5 )/2 Michael Creutz BNL 24
Message for continuum QCD: physicalθcan be moved around placed on any one flavor at will Θ can be entirely moved into the top quark phase top quark properties relevant to low energy physics! decoupling theorems don t apply non-perturbatively Michael Creutz BNL 25
Summary Excitations on a spinless Dirac sea can carry spin required by relativistic form of spectrum Close connections with chiral symmetry mass topologically protected doublers required entwined with the CP violating parameter Θ Michael Creutz BNL 26