Moebius DW Fermions & Ward-Takahashi identities * Brookhaven Nat’l Lab. Ides of March, 2007 * RCB, Hartmut Neff and Kostas Orginos hep-lat/ & hep-lat/0703XXX * RCB, Hartmut Neff and Kostas Orginos hep-lat/ & 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(M 5 ) Two steps

 L [x]| at L s = 16 (polar) vs 8 (Zolotarev)

Mobius generalization of Shamir/Borici Shamir: b 5 = a 5, c 5 = 0Borici: b 5 = c 5 = a 5 s = 1s = 2s = 3s = L_s x x -  x +  D-D- D+D+ D-D-

Modified Even/Odd  4-d Checkerboard b5b5 a5a5 1+  5 1-  5 b5b5 x x -  x +  s = 1s = 2s = 3s = L_s

Even/Odd Partition of Matrix On 16 3 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 I ee and I oo SHOULD be simple to invert

DW Construction: Shamir vs Borici

Moebius Generalization Parameters: M 5, a 5 = b 5 – c 5 and scale: a = b 5 + c 5 Since Moebius is an new (scaled) polar algorithm

16 3 x 32 Gauge  = 6 & m  = 0.44

Domain Wall Implementation L s £ L s DW Matrix: ( need s dependence for Chiu’s Zolotarev: b 5 (s) + c 5 (s) =  (s), b 5 (s) – c 5 (s) = a 5 )

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/ )

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,1 y,1y

Split Screen Correlators: 5-d Vector => 4-d Axial s = 1s = 2s = Ms = L_s qLqL qRqR QLQL QRQR qLqL qRqR QRQR QLQL LEFT RIGHT

Bulk to Boundary Propagators where s = M plane (See Kikukawa and Noguchi, hep-lat/ ) y y

Ward Takashi: DW => Overlap y 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 m res dependence on  & L s  ( ) has negligible dependence on  and L s (Parameterize and fit m res data)