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Moebius DW Fermions & Ward-Takahashi identities *. Richard C. Brower. Brookhaven Nat’l Lab. Ides of March, 2007. * RCB, Hartmut Neff and Kostas Orginos hep-lat/0409118 & hep-lat/0703XXX. Need to implement Overlap Operator (aka Ginsparg Wilson Relation). Two steps. - PowerPoint PPT Presentation
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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
*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)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
b5
a5
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 PropagatorsBulk to Boundary Propagators
where s = M plane
(See Kikukawa and Noguchi, hep-lat/99902022)
y
y
Ward Takashi: DW => Overlapy 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)
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