Lattice Fermions A

Lattice Fermions a

Michael Creutz BNL 1 Lattice Fermions

Consider some lattice Dirac operator D

† • assume gamma five hermiticity γ5Dγ5 = 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 2 Then by construction

[K,γ5]+ = 0

[M,γ5]− = 0

On a lattice everything is finite; so Trγ5 = 0

M → eiθγ5 M is an exact symmetry of the determinant

|K + M| = |eiγ5θ/2(K + M)eiγ5θ/2| = |K + eiθγ5 M|

Where did the anomaly hide?

Michael Creutz BNL 3 This must be a flavored chiral symmetry

All lattice actions bring in extra structure

Naive and staggered fermions have doublers

• half use γ5 and half −γ5 • the naive chiral symmetry is actually flavored

Wilson and overlap fermions • M is not a constant • heavy states appear to cancel the anomaly • chiral symmetry modified by their mass

Michael Creutz BNL 4 Continuum free fermion action density

ψDψ = ψ(∂/ + m)ψ

in momentum space

ψ(ip/ + m)ψ.

D has both Hermitean and anti-Hermitean parts

• Hermitian mass term is a constant

• this won't be the case on the lattice

Michael Creutz BNL 5 Lattice transcription replaces p with trigonometric functions • Fourier transform ˜ −iqj π dq iqj ˜ φq = j e φj φj = −π 2π e φq ∗ ∗ π dq ˜∗ ˜ • φj+1φj − φj φj+1 = −2i −π 2π sin(q)φ (q)φ(q) Naive lattice fermions

• replace ∂µ with nearest neighbor differences

i ψ a µ γµ sin(pµa)+ m ψ • at small p goes to desired ψ(iγµpµ + m)ψ

But this also has low energy excitations for pµ ∼ π/a • we have 24 = 16 ``doublers''

Michael Creutz BNL 6 Wilson: Add a momentum dependent mass

1 ψDW ψ = ψ a µ(iγµ sin(pµa) + 1 − cos(pµa)) + m ψ. 1 • for small momentum a (1 − cos(pµa)) = O(a)

• for momentum components near π the eigenvalues are of order 1/a

Eigenvalues for the free theory at

i 2 1 λ = ± a µ sin (pµa)+ a µ(1 − cos(pµa)) + m. • both real and imaginary parts even at m = 0

Michael Creutz BNL 7 The eigenvalues form a set of ``nested circles''

Im λ λ Re m

Notes • m ↔−m not a symmetry

• essential for quantum theory with Nf odd

• chiral symmetry broken: [D,γ5]+ = 0 • m can get an additive renormalization

Michael Creutz BNL 8 The Nielsen-Ninomiya theorem

Doubling closely tied to topology in momentum space • Consider the gamma matrix convention 0 σ γ = σ ⊗ σ = 1 σ 0 0 −i γ = σ ⊗ I = 0 2 i 0 1 0 γ = σ ⊗ I = 5 3 0 −1

Consider some anti-Hermitean Dirac operator D anti-commuting with γ5 † D = −D = −γ5Dγ5.

Go to momentum space: D(p) a 4 × 4 matrix function of pµ

Michael Creutz BNL 9 The most general form is

0 z(p) D(p)= −z∗(p) 0

• where z(p) is a quaternion

z(p)= z0(p)+ iσ z(p).

Any chirally symmetric Dirac operator

• maps momentum space onto the space of quaternions

Michael Creutz BNL 10 Dirac equation expands D(p) around a zero

• D(p) ≃ ip/ = iγµpµ

Consider sphere of constant |p| surrounding a zero

• this maps z non-trivially around the origin in quaternion space z 0

Michael Creutz BNL 11 Non-trivial mapping makes the zeros robust

E E

p p

allowed forbidden

• masslessness is protected

Michael Creutz BNL 12 Momentum space periodic over a Brilloin zone −π ≤ pµ < π • must have z(p)= z(p + 2πn) Any wrapping must unwrap elswhere p 2 π

2π/3

p 1 −π −2π/3 2π/3 π

−2π/3

−π

Naive 16 doublers divide into two groups of 8 • with opposite mappings An exact chiral lattice theory must have an even number of species

Michael Creutz BNL 13 Minimal doubling

Several local chiral lattice actions with Nf = 2 flavors are known

• symmetry preserves multiplicative mass renormalization

Known variations break hyper-cubic symmetry

• introduces anisotripic counter terms

A single local field can create two species

Michael Creutz BNL 14 Karsten Wilczek example

• replace Wilson term with imaginary chemical potential

• the free momentum space Dirac operator

3 iγ4 4 D(p)= i i=1 γi sin(pi)+ sin(α) cos(α) + 3 − µ=1 cos(pµ) • exact chiral symmetry: [D,γ5]+ = 0

Propagator has two poles, at p = 0, p4 = ±α

• parameter α allows adjusting the relative pole positions

• original form used α = π/2, cos(α) = 0

Michael Creutz BNL 15 Karsten Wilzcek action maintains one exact chiral symmetry

[D,γ5]+ = 0

The two species are not equivalent • they have opposite chirality • expand the propagator around the poles

around p4 =+α uses the usual gamma matrices ′ −1 p4 = −α uses γµ = Γ γµΓ

for this action Γ= iγ4γ5 ′ γ5 = −γ5 • exact chiral symmetry is ``flavored''

like continuum τ3γ5

Michael Creutz BNL 16 Inserting gauge fields

3 δi,j+e − δi,j−e D =U γ µ µ ij ij i 2 µ=1 4 iγ δi,j+e + δi,j−e + 4 (cos(α)+3)δ − U µ µ . sin(α) ij ij 2 µ=1

Broken hypercubic symmetry can induce asymmetry • renormalization of α • renormalization of temporal hopping • renormalization of time-like plaquettes

Issues with controlling these counter-terms await simulations

Michael Creutz BNL 17 A single field ψ can create either of the two species

Can separate them with point splitting

Consider for the free theory

1 sin(q + α) u(q)= 1+ 4 ψ(q + αe ) 2 sin(α) 4 1 sin(q − α) d(q)= Γ 1 − 4 ψ(q − αe ) 2 sin(α) 4

• here Γ= iγ4γ5 for the Karsten/Wilczek action accounts for two species useing different gammas • construction not unique

Michael Creutz BNL 18 Inserting gauge field factors in position space

1 U ψ − U ψ u = eiαx4 ψ + i x,x−e4 x−e4 x,x+e4 x+e4 x 2 x 2sin(α) 1 U ψ − U ψ d = Γe−iαx4 ψ − i x,x−e4 x−e4 x,x+e4 x+e4 . x 2 x 2sin(α)

• phases remove oscillations from poles at non-zero momentum

Michael Creutz BNL 19 Domain wall and overlap fermions

Wilson fermions with K > Kc have low energy surface modes • naturally chiral • mixing through tunnelling between walls • example of a ``topological insulator'' robust conductiity only on surfaces Use this for ``domain wall'' fermions • work on a 5-d lattice of finite size 0 ≤ x5 < ls • identify surface modes with physical quarks • bulk modes go to infinite mass in continuum limit

Michael Creutz BNL 20 Overlap fermion limit • drive bulk modes to large energy

• take ls →∞ • name from ground state ``overlap'' of 5d transfer matrices

Can be reformulated directly in four dimensions • possess an order a modified chiral symmetry ψ −→ eiθγ5 ψ ψ −→ ψeiθ(1−aD)γ5 † • also maintain γ5Dγ5 = D

Note the asymmetric treament of ψ and ψ

Michael Creutz BNL 21 Invariance of ψDψ implies

Dγ5 = −γ5D + aDγ5D = −γˆ5D

• where γˆ5 ≡ (1 − aD)γ5.

• this is the ``Ginsparg-Wilson relation''

• naive anticommutation corrected by O(a)

The GW relation is equivalent to the unitarity of

V = 1 − aD

GW −→ V †V = 1

Michael Creutz BNL 22 Neuberger: construct V via unitarization • start with an undoubled Dirac operator

• i.e. the Wilson operator Dw † −1/2 V = −Dw(DwDw) . † −1/2 • to define (DwDw) † 1) diagonalize DwDw 2) take the square root of the eigenvalues

3) undo the diagonalizing unitary transformation.

Given V , the overlap operator is D = (1 − V )/a.

Michael Creutz BNL 23 Im λ Im λ

λ λ Re Re m m 0 2

Properties of the overlap • computationally demanding • ``normal:'' [D†, D] = 0, unlike Wilson † • γ5Dγ5 = D • a modified exact chiral symmetry ψ → eiθγ5 ψ ψ → ψeiθγˆ5

• where γˆ5 = Vγ5.

Michael Creutz BNL 24 2 As with γ5, γˆ5 is Hermitean and γˆ = 1

• all eigenvalues are ±1

• defines an index

1 γ5+ˆγ5 ν = 2 Trˆγ5 = Tr 2

D has ν exact zero eigenvalues

• agrees with continuum index for smooth fields

• 1/2 compensates real modes on opposite side of the unitarity circle

Michael Creutz BNL 25 Some issues • The ``kernel'' projected for the overlap is somewhat arbitrary

with Dw, dependence on the hopping parameter

need K > Kc so one species projects out need K below doubler masses or too many massless fermions

• if K < Kc still satisfy GW but no massless particles

Im λ Im λ

λ λ Re Re m 0 2 Nf = 0

Im λ Im λ

λ λ Re Re m 0 2 N5 = 5 GW does NOT require massless Goldstone bosons

Michael Creutz BNL 26 Issues continued

• ν can depend on choice of K

depends on eigenvalue density in first circle