Moebius DW Fermions&

Ward-Takahashi identities*

Brookhaven Nat’l Lab. Ides of March, 2007

*RCB, Hartmut Neff and Kostas Orginoshep-lat/0409118 & hep-lat/0703XXX

Richard C. Brower

Need to implement Overlap Operator(aka Ginsparg Wilson Relation)

Algorithm: Choose Rational approx:

L[x] ' x/|x|

Physics: Choose 4-d “kernel”:

H ´ 5 D(M5)

Two steps

L[x]| at Ls = 16 (polar) vs 8 (Zolotarev)

Mobius generalization of Shamir/Borici

Shamir: b5 = a5, c5 = 0 Borici: b5 = c5 = a5

s = 1 s = 2 s = 3 s = L_s

x

x -

x +

D-

D+D-

Modified Even/Odd 4-d Checkerboard

b5a5

1+51-5

b5

x

x -

x +

s = 1 s = 2 s = 3 s = L_s

Even/Odd Partition of Matrix

On 163 x 32, = 6.0 lattice• For Shamir: Both 4-d & 5-d Even/Odd give » 2.7 speed up.• For Borici/Moebius: Even/Odd gives » 2.7 speed up

Iee and Ioo SHOULD be simple to invert

DW Construction: Shamir vs Borici

Moebius Generalization

Parameters: M5 , a5 = b5 – c5 and scale: a = b5 + c5

Since Moebius is an new (scaled) polar algorithm

163 x 32 Gauge Lattice @ = 6 & m = 0.44

Domain Wall ImplementationLs £ Ls DW Matrix:

( need s dependence for Chiu’s Zolotarev:b5(s) + c5(s) = (s), b5(s) – c5(s) = a5)

• To get all the nice identities for Borici, Chiu and Mobius

Generalized 5 Hermiticity and All That

G-W error operator:

Chiral violation for Overlap Action (Kikukawa & Noguchi hep-lat/9902022)

Edwards & Heller use “Standard” UDL decomposition

Step #1: Prepare the Pivots by Permute Columns

Step #3 Back substitution to get L matrix

Step #2: Do Gaussian Elimination to get U matrix

where

LUD =>

DW/Overlap Equivalence:

note: Standard approach

where

Application of DW/overlap equ to currents

z z,1y,1

y

Split Screen Correlators: 5-d Vector => 4-d Axial

s = 1 s = 2 s = M s = L_s

qL

qR

QL

QRqL

qR

QR

QL

LEFT RIGHT

Bulk to Boundary Propagators

where s = M plane

(See Kikukawa and Noguchi, hep-lat/99902022)

y

y

Ward Takashi: DW => Overlapy

implies

where

( y For single current disconnected diagram gives anomaly)

Measuring the Operator Ls

(use Plateau region away from sources)

Sum over t Measure Matrix element of L operator

| > in the Eigen basis of H = 5 D(-M)

where

Derivation:

Model for mres dependence on & Ls

() has negligible dependence on and Ls

(Parameterize and fit mres data)