New Staggered-type Fermions on the Lattice:
A ReviewDavid Adams
Division of Mathematical Sciences
Nanyang Technological University, Singapore

Outline
Introduction

Outline
Introduction
Traditional approaches: Wilson fermions & staggered fermionsOutline
Introduction
Traditional approaches: Wilson fermions & staggered fermionsWilson Terms for Staggered FermionsOutline
Introduction
Traditional approaches: Wilson fermions & staggered fermionsWilson Terms for Staggered Fermions
Properties of Staggered Wilson FermionsOutline
Introduction
Traditional approaches: Wilson fermions & staggered fermionsWilson Terms for Staggered Fermions
Properties of Staggered Wilson FermionsIndex Theorem

Outline
Introduction
Traditional approaches: Wilson fermions & staggered fermionsWilson Terms for Staggered Fermions
Properties of Staggered Wilson FermionsIndex Theorem
Example of a Meson PropagatorOutline
Introduction
Traditional approaches: Wilson fermions & staggered fermionsWilson Terms for Staggered Fermions
Properties of Staggered Wilson FermionsIndex Theorem
Example of a Meson Propagator
Summary

Outline
Introduction
Traditional approaches: Wilson fermions & staggered fermionsWilson Terms for Staggered Fermions
Properties of Staggered Wilson FermionsIndex Theorem
Example of a Meson Propagator
SummaryIntroduction
Will discuss staggered versions of Wilson fermions, domain wall
fermions and overlap fermions on the lattice.
They are theoretically novel, and the hope is that they might also
be computationally more efficent than the usual Wilson-based
fermions.
Background:
Originated from an attempt to identify and understand thewould-be zero-modes and index of the staggered Dirac operator,and construct overlap fermions from staggered fermions.
[D.A., PRL (2010), PLB (2011)]
Will review the developments and recent results discussed at thisworkshop.

Outline
Introduction
Traditional approaches: Wilson fermions & staggered fermionsWilson Terms for Staggered Fermions
Properties of Staggered Wilson FermionsIndex Theorem
Example of a Meson Propagator
SummaryTraditional Approaches to Lattice Fermions◮ Start from naive discretization of Dirac operator:
1
Dnaive = γµ ∇µ = γµ 2a(Tµ+ − Tµ− )whereTµ+ ψ(x) = Uµ (x)ψ(x + aµ̂)

,Tµ− = (Tµ+ )−1 .

◮ Free field momentum rep isD̂naive (p) = iγµ 1a sin(apµ )→ D̂naive (p) = 0 has 16 solutions for p ∈ (− πa , πa ]4 .
→ 16 lattice fermion species → 15 spurious “doublers”.

Traditional Approaches (continued)
Wilson fermions: Add a term to give mass ∼ 1a to the 15spurious species:
DW = γµ ∇µ + a 2r ∆→ D̂W (p) = iγµ 1a sin(apµ ) + ar∆ = lattice Laplace op.
Pν (1 − cos(apν ))→ D̂W (p) = 0 only has one solution: p = 0.
Disadvantages:◮ The continuum chiral symm
Wilson term a 2r ∆.

{D , γ5 } = 0 is broken by the◮ O(a2 ) discretization error of naive fermion becomes O(a).

Traditional Approaches (continued)
Staggered fermions: The 16 species of Dnaive = γµ ∇µ can be
reduced to 4 species via spin-diagonalization:Λψ(x) = γ1n1 · · · γ4n4 ψ(x)x = a(n1 , n2 , n3 , n4 )givesΛ−1 (γµ ∇µ )Λ = Dst 1
where
Dst = ηµ ∇µ,ηµ ψ(x) = (−1)n1 +···+nµ−1 ψ(x)Note: Dst is a scalar operator.
⇒ Naive lattice fermion ≃ 4 copies of staggered fermion
described by Dst .

About Staggered Fermions◮ Staggered lattice fermion field χ(x) is scalar (no spinorindices).

About Staggered Fermions◮ Staggered lattice fermion field χ(x) is scalar (no spinorindices).
◮ Describes 4 Dirac fermion species (i.e. 4 flavors).

Spin–flavor interpretation is tricky.

About Staggered Fermions◮ Staggered lattice fermion field χ(x) is scalar (no spinorindices).
◮ Describes 4 Dirac fermion species (i.e. 4 flavors).

Spin–flavor interpretation is tricky.
◮ →Disadvantage: have to live with the 4 copies (“tastes”) ofeach physical quark.

About Staggered Fermions◮ Staggered lattice fermion field χ(x) is scalar (no spinorindices).
◮ Describes 4 Dirac fermion species (i.e. 4 flavors).

Spin–flavor interpretation is tricky.
◮ →Disadvantage: have to live with the 4 copies (“tastes”) ofeach physical quark.

◮ Advantage: One of the flavored chiral symms holds exactly:{Dst , Γ55 } = 0 whereΓ55 χ(x) = (−1)n1 +n2 +n3 +n4 χ(x)Interpretation: Γ55 = γ5 ⊗ γ5 on spin⊗flavor.

About Staggered Fermions◮ Staggered lattice fermion field χ(x) is scalar (no spinorindices).
◮ Describes 4 Dirac fermion species (i.e. 4 flavors).

Spin–flavor interpretation is tricky.
◮ →Disadvantage: have to live with the 4 copies (“tastes”) ofeach physical quark.

◮ Advantage: One of the flavored chiral symms holds exactly:{Dst , Γ55 } = 0 whereΓ55 χ(x) = (−1)n1 +n2 +n3 +n4 χ(x)Interpretation: Γ55 = γ5 ⊗ γ5 on spin⊗flavor.
◮ Discretization error remains at O(a2 ).
◮ Various exact lattice spacetime symmetries.

About Staggered Fermions◮ Staggered lattice fermion field χ(x) is scalar (no spinorindices).
◮ Describes 4 Dirac fermion species (i.e. 4 flavors).

Spin–flavor interpretation is tricky.
◮ →Disadvantage: have to live with the 4 copies (“tastes”) ofeach physical quark.

◮ Advantage: One of the flavored chiral symms holds exactly:{Dst , Γ55 } = 0 whereΓ55 χ(x) = (−1)n1 +n2 +n3 +n4 χ(x)Interpretation: Γ55 = γ5 ⊗ γ5 on spin⊗flavor.
◮ Discretization error remains at O(a2 ).
◮ Various exact lattice spacetime symmetries.
◮ Computationally efficient. In particular, improved versions getclose to the continuum limit with manageable cost.

About Staggered Fermions◮ Staggered lattice fermion field χ(x) is scalar (no spinorindices).
◮ Describes 4 Dirac fermion species (i.e. 4 flavors).

Spin–flavor interpretation is tricky.
◮ →Disadvantage: have to live with the 4 copies (“tastes”) ofeach physical quark.

◮ Advantage: One of the flavored chiral symms holds exactly:{Dst , Γ55 } = 0 whereΓ55 χ(x) = (−1)n1 +n2 +n3 +n4 χ(x)Interpretation: Γ55 = γ5 ⊗ γ5 on spin⊗flavor.
◮ Discretization error remains at O(a2 ).
◮ Various exact lattice spacetime symmetries.
◮ Computationally efficient. In particular, improved versions getclose to the continuum limit with manageable cost.

Spin–Flavor Interpretation of Staggered Fermions◮ Momentum space approach:π 3π 4, 2a ] written asp ∈ [− 2ap = q+ πa A ,ππ 4q ∈ [− 2a, 2a] , A = (A1 , . . . , A4 ) , Aµ ∈ {0, 1}→ χ̂(p) = χ̂(q + πa A) ≡ χ̂A (a).
→ Free field momentum rep of Dst = γµ ∇µ has the formΓ̂µ ai sin(aqµ )where Γ̂µ = (Γ̂µ )AB are 16 × 16 matrices giving a 16-dim repof Dirac algebra.
→ decomposes into 4 copies of 4-dim rep: the 4 flavors.
◮ There is also a free field coordinate space approach.

Outline
Introduction
Traditional approaches: Wilson fermions & staggered fermionsWilson Terms for Staggered Fermions
Properties of Staggered Wilson FermionsIndex Theorem
Example of a Meson Propagator
Summary

How about a Wilson term for staggered fermions?
Goal:
Reduce number of fermion species from 4 to 1 (or 2).
Then will have fewer “doubler” species than in usual Wilson case:3 (or 2) versus 15.
→ More efficient than usual Wilson fermions.

How about a Wilson term for staggered fermions?
Goal:
Reduce number of fermion species from 4 to 1 (or 2).
Then will have fewer “doubler” species than in usual Wilson case:3 (or 2) versus 15.
→ More efficient than usual Wilson fermions.
Natural approach is to add a momentum-dependent “flavored”
mass term Wst to staggered Dirac operator:
Dst → Dst + 1a Wst
Zero-eigenmodes for Wst → physical fermion species
Nonzero eigenmodes for Wst → doubler species with mass ∼ 1/aHow about a Wilson term for staggered fermions?
Goal:
Reduce number of fermion species from 4 to 1 (or 2).
Then will have fewer “doubler” species than in usual Wilson case:3 (or 2) versus 15.
→ More efficient than usual Wilson fermions.
Natural approach is to add a momentum-dependent “flavored”
mass term Wst to staggered Dirac operator:
Dst → Dst + 1a Wst
Zero-eigenmodes for Wst → physical fermion species
Nonzero eigenmodes for Wst → doubler species with mass ∼ 1/aTo avoid complex fermion det, require Wst† = Γ55 Wst Γ55 .

Flavored Mass Terms for Staggered Fermions
The possible flavoured mass terms for staggered fermions were
found long ago [Golterman & Smit (1984)].
Ingredients:
Γ55 χ(n) = (−1)n1 +n2 +n3 +n4 χ(n)◮

Γ55 ≃ γ5 ⊗ γ5◮

Γ5 ≃ γ5 ⊗ 1 + O(a2 )Γ 5 = η5 C

where
η5 = η1 η2 η3 η4,C = (C1 C2 C3 C4 )sym

η5 χ(n) = (−1)n1 +n3 χ(n),1Cµ = 2a(Tµ+ + Tµ− )Recall Tµ+ χ(n) = Uµ (n)χ(n + µ̂) is parallel transporter.

Wilson Terms for Staggered Fermions
The most general flavored mass term satisfying Γ55 -hermiticity isP
Wst = c1 + µ<ν cµν Mµν + c5 M5where the 2-link operators Mµν and 4-link operator M5 are given byMµν= iηµν Cµν,Cµν = 12 (Cµ Cν + Cν Cµ )M5 = η5 Γ55 Γ5
Spin-flavor interpretations:Mµν ∼ 1 ⊗ iγµ γν

,M5 ∼ 1 ⊗ γ 5

For staggered Wilson term, choose c, the cµν ’s and c5 so that Wst
has 1 (or 2) zero-eigenmodes.

Wilson Terms for Staggered Fermions (cont’d)This is quite an obvious possibility, so why didn’t Golterman &Smit or others already do it a long time ago?
Answer (my guess):
1. Concern about breaking lattice rotation invariance with theMµν ’s.
2. Loses 2 key advantageous features of staggered fermions:◮

Exact flavored chiral symm {Dst , Γ55 } = 0 is broken.

◮O(a2 ) discretization error becomes O(a).

3. Compared to usual Wilson fermion, is the gain in efficiency (ifit is even realized) enough to justify the more complicatedspin-flavor structure?
In the old days, efficiency was not a pressing concern. It is more sonow.

Why is staggered-Wilson more interesting now?
◮ Realistic unquenched Lattice QCD simulations are now

possible.
→ Much efforts to find “improved” formulations to get closer
to the continuum limit and chiral limit without an excessive
increase in computing cost.
Can staggered-Wilson do significantly better than usual
Wilson for this?
◮ Chirally improved lattice fermion formulations have beenfound: domain wall fermions and overlap fermions.
– Built from Wilson fermions.
– Attractive theoretical properties but computationally very
expensive⇒ Look for more computationally efficient versions of these.
Can staggered-Wilson give a more efficient version of domain
wall and overlap fermions?

A 2-flavor staggered Wilson term
Work on the staggered fermion index and a related staggered
overlap fermion construction [D.A. PRL (2010), PLB (2011)] led
to staggered Wilson fermion with Wilson term constructed withthe flavored mass term M5 = Γ55 Γ5 :Wst = 1a (1 − Γ55 Γ5 )Recall Γ55 Γ5 ≡ 1 ⊗ γ5 + O(a2 )Decompose the 4 Dirac fermion spaciesof the staggered fermion
into 2 species with positive flavor-chirality under 1 ⊗ γ5 and 2species with negative flavor-chirality.
Then Wst keeps the 2 positive flavor-chirality species as thephysical fermions and gives mass ∼ 1/a to the negative
flavor-chirality species; they become the “doublers”.
→ Get 2-flavor staggered-Wilson fermion on the lattice.

Free field spectrum of 2-flavor staggered-Wilson[P. de Forcrand, Lattice 2010 conf.]

Green: Eigenvalues of free field DsW (staggered Wilson)Blue: Eigenvalues of free field DW (usual Wilson).
2Im[λ]

10-1-20246Re[λ]→ Spectrum of staggered Wilson is sensible and contains
much less junk than spectrum of usual Wilson.

Symmetries of 2-flavor staggered-Wilson
Classical action is χ̄DsW χ whereDsW = Dst + ra0 (1 − Γ55 Γ5 ) + m0r0 , m0 : bare parameters.
It is invariant under lattice rotations and all the symmetries of theoriginal staggered fermion, except the “shift transformations” – forthese haveχ̄Γ55 Γ5 χ → −χ̄Γ55 Γ5 χTurns out only one new counter-term is possible: {Γ55 Γ5 , Dst }.

Symmetries of 2-flavor staggered-Wilson (cont’d)→ The quantum effective action can be put in the form χ̄DsW χwhereDsW = (1 + cΓ55 Γ5 )Dst + ar (1 − Γ55 Γ5 ) + mandm = m0 +c′aNote Γ55 Γ5 Dst ≡ (1 ⊗ γ5 )Dst which coincides with Dst on thephysical species.
→ Can approach massless (chiral) limit by tuning bare mass m0 :m0 → −c′aThis is basically the same situation as for usual Wilson fermions.

2-flavor staggered-Wilson (cont’d)◮ Should check that a massless (chiral) limit can be approachedby calculating the “pion” mass as a function of the bare quark
mass in a Lattice QCD simulation.
◮ Also, do same with usual Wilson fermions and see if

staggered-Wilson is more chiral, i.e. can reach a smaller pionmass than usual Wilson.
Simulation details: quenched simulation with 50 configs at β = 6.0
on two lattices: 123 × 32 and 163 × 32.
Done by Andriy Petrashyk, following instructions from DanielNogradi.

Pion mass plots0.3

123x32163x32Wilson
StW

0.25

0.2
mπ22.2813*mq + 1.8540

0.150.1

3.1322*mq + 2.3338
0.05
-0.77

-0.76

-0.75

-0.74

-0.73

-0.72
mq-0.71-0.7-0.69-0.68

2-flavor staggered-Wilson: spectrum calculations◮ The spectrum of the staggered-Wilson Dirac operator in a
typical quenched β = 6 background on an 84 lattice waspresented by Ph. de Forcrand.
It doesn’t look good, but usual Wilson spectrum was notpresented for comparison.
◮ The spectrum looks even worse on 44 lattice at β = 5.6(results of S. Durr, presented at the workshop). Comparison
with Wilson spectrum was also made in this case andstaggered-Wilson looks worse.
◮ It would be interesting to compare staggered-Wilson and usual
Wilson spectra in quenched β = 6 background on 163 × 32

lattice– the pion mass results suggest staggered-Wilson may bebetter in that case.

1-flavor staggered-Wilson[C. Hoelbling, PLB (2011)]
Consider a staggered Wilson term built from the Mµν flavoredmass terms:
WstH = 1a (M12 + M13 + M14 + M23 + M24 + M34 )(or variants with different sign combinations).
This splits the flavor degeneracy to provide a 1-flavor
staggered-Wilson theory with Dirac operator
HDsW= Dst + 1a (WstH + 1)

Free field spectrum for 1-flavor staggered-Wilson[C. Hoelbling, this workshop.]

Blue: 1-flavor staggered-Wilson
Purple: 2-flavor staggered-Wilson
Yellow: Usual Wilson.

1-flavor staggered-Wilson: symmetries
It maintains all staggered fermion symmetries except for the shifttransformations:these change χ̄Mµν χ → −χ̄Mµν χ if the shift is along the µ− or
ν− axes.
However, lattice rotation symmetry is broken.
E.g. under R (12) have
WstH →1a (M12 − M13 + M14 + M23 − M24 + M34 )But still have a residual lattice rotation symmetry:
WstH is invariant under double rotations R (µν) R (σρ) when µ, ν, σ, ρare all different.

Is 1-flavor staggered-Wilson viable?
◮ Spacetime rotation symmetry along with gauge invariance isessential fr renormalizability of continuum QCD. –excludes
new counterterms from arising.
◮ In lattice QCD the rotation symmetry is broken down to thediscrete subgroup of hypercubic lattice rotations.
Miraculously, this is still enough to exclude new counterterms
and maintain renormalizability [T. Reisz].
◮ For 1-flavor staggered-Wilson the spacetime rotation

symmetry is broken even further, down to the double rotations
R (µν) R (σρ) for µ, ν, σ, ρ all different.
Can we hope for a further miracle?
(Doubtful IMHO)Outline
Introduction
Traditional approaches: Wilson fermions & staggered fermionsWilson Terms for Staggered Fermions
Properties of Staggered Wilson FermionsIndex Theorem
Example of a Meson PropagatorSummaryIndex via Spectral Flow
Spectral flow of H(m) in a U(1) background with Q = 1 on
12 × 12 lattice:m on horizontal axis; eigenvalues of H(m) on vertical axis
221100−1−1−2−1012

staggered Wilson
HsW (m) = Γ55 (DsW − m)

3−2−101234usual Wilson
HW (m) = γ5 (DW − m)

5Index via Spectral Flow (continued)
Spectral flow of H(m) in another U(1) background with Q = −2on 12 × 12 lattice:221100−1−1−2−1012

staggered Wilson
HsW (m) = Γ55 (DsW − m)

3−2−101234

usual Wilson
HW (m) = γ5 (DW − m)

5