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![Page 1: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/1.jpg)
Topology conserving actions and the overlap Topology conserving actions and the overlap Dirac operatorDirac operator
(hep-lat/0510116) (hep-lat/0510116) Hidenori FukayaYukawa Institute, Kyoto Univ.
Collaboration with S.Hashimoto (KEK,Sokendai), T.Hirohashi (Kyoto Univ.),H.Matsufuru(KEK), K.Ogawa(Sokendai) and T.Onogi(YITP)
![Page 2: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/2.jpg)
Contents
1. Introduction2. The overlap fermion and topology3. Lattice simulations4. Results5. Conclusion and outlook
![Page 3: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/3.jpg)
Lattice regularization of the gauge theory is a very powerful tool to analyze strong coupling regime but it spoils a lot of symmetries…
Translational symmetry Lorentz invariance Chiral symmetry or topology Supersymmetry…
1. Introduction
![Page 4: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/4.jpg)
Nielsen-Ninomiya theorem
Any local Dirac operator satisfying chiral symmetry has unphysical poles (doublers).
Example - free fermion – Continuum has no double r . Lattice
has unphysical poles at . Wilson fermion
Doublers are decoupled but no chiral symmetry.
Nucl.Phys.B185,20 (‘81),Nucl.Phys.B193,173 (‘81)
![Page 5: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/5.jpg)
The Ginsparg-Wilson relation
The Neuberger’s overlap operator:
satisfying the Ginsparg-Wilson relation:
realizes ‘modified’ exact chiral symmetry on the lattice;the action is invariant under
NOTE Expansion in Wilson Dirac operator ⇒ No doubler. Fermion measure is not invariant;
⇒ chiral anomaly, index theorem
Phys.Rev.D25,2649(‘82)
Phys.Lett.B417,141(‘98)
![Page 6: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/6.jpg)
The overlap Dirac operator
The overlap operator
becomes ill-defined when
These zero-modes are lattice artifacts. (excluded in the continuum limit.)
Locality may be lost. (no zero-modes ⇒ guaranteed.) The boundary of topological sectors. The determinant is also non-smooth
⇒ numerical cost is expensive.
![Page 7: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/7.jpg)
Topology conserving actions
can be achieved by
The “admissibility” condition
The determinant (The negative mass Wilson fermion)
Details are in the next section…
![Page 8: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/8.jpg)
Our goals
Motivation : Exactly chiral symmetric Lattice QCD with the overlap Dirac operator.
Problem : should be excluded for
sound construction of quantum field theory (Determinant should be a smooth function )
numerical cost down
Solution ? : Topology conserving actions ?
Practically feasible? (Small O(a) errors? Perturbation?) Topology is really conserved? Numerical costs ? Let’s try !
c.f. W.Bietenholz et al. hep-lat/0511016.
![Page 9: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/9.jpg)
02/a 4/a 6/a
Eigenvalue distribution of Dirac operators
2. The overlap fermion and topology
continuum
(massive)
m
1/a
-1/a
![Page 10: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/10.jpg)
Wilson fermion
Eigenvalue distribution of Dirac operators
2. The overlap fermion and topology
1/a
-1/a
naïve fermion
16 lines
02/a 4/a 6/a
(massive)
m
• Doublers are massive.
• m is not well-defined.
![Page 11: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/11.jpg)
Eigenvalue distribution of Dirac operators
2. The overlap fermion and topology
1/a
-1/a
02/a 4/a 6/a
The overlap fermion
• D is smooth except for .
![Page 12: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/12.jpg)
Eigenvalue distribution of Dirac operators
2. The overlap fermion and topology
1/a
-1/a
02/a 4/a 6/a
The overlap fermion(massive)
m
• Doublers are massive.
• m is well-defined.
![Page 13: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/13.jpg)
Eigenvalue distribution of Dirac operators
2. The overlap fermion and topology
1/a
-1/a
02/a 4/a 6/a
The overlap fermion
• Topology boundary.
• Locality may be lost.
• Large simulation cost.
![Page 14: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/14.jpg)
The topology (index) changes
2. The overlap fermion and topology
1/a
-1/a
02/a 4/a 6/a
The complex modes make pairs
The real modes are chiral eigenstates.
![Page 15: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/15.jpg)
The locality P.Hernandez et al. (Nucl.Phys.B552,363 (1999)) proved
where A and ρ are constants. Numerical cost
In the polynomial approximation for D
The discontinuity of the determinant requires reflection/refraction (Fodor et al. JHEP0408:003,2004)
2. The overlap fermion and topology
![Page 16: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/16.jpg)
The topology conserving gauge action
generates configurations satisfying the “admissibility” bound:NOTE: The effect of ε is O(a4) and the positivity is restored as
ε/a4 → ∞ . Hw > 0 if ε < 1/20.49, but it’ s too small…
2. The overlap fermion and topology
M.Creutz, Phys.Rev.D70,091501(‘04)
M.Luescher,Nucl.Phys.B568,162 (‘00)
Let’s try larger ε.
![Page 17: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/17.jpg)
The negative mass Wilson fermion
would also suppress the topology changes. would not affect the low-energy physics in principle. but may practically cause a large scaling violation.
Twisted mass ghosts may be useful…
2. The overlap fermion and topology
![Page 18: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/18.jpg)
How to sum up the different topological sectors
2. The overlap fermion and topology
⇒ We need ..
![Page 19: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/19.jpg)
How to sum up the different topological sectors
With an assumption,The ration can be given by the topological
susceptibility,
if it has small Q and V’ dependences. Parallel tempering + Fodor method may also be useful.
2. The overlap fermion and topology
V’
Z.Fodor et al. hep-lat/0510117
![Page 20: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/20.jpg)
In this talk,
Topology conserving gauge action (quenched)
Negative mass Wilson fermion
Future works …
Summation of different topology Dynamical overlap fermion at fixed topology
3. Lattice simulations
![Page 21: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/21.jpg)
The simulations were done on the Alpha work station at YITP and SX-5 at RCNP.
Topology conserving gauge action (quenched)
with 1/ε= 1.0, 2/3, 0.0 (=plaquette action) . Algorithm: The standard HMC method. Lattice size : 124,164,204 . 1 trajectory = 20 - 40 molecular dynamics steps
with stepsize Δτ= 0.01 - 0.02.
3. Lattice simulationssize 1/ε β Δτ Nmds acceptanc
ePlaquette
124 1.0 1.0 0.01 40 89% 0.539127(9)1.2 0.01 40 90% 0.566429(6)1.3 0.01 40 90% 0.578405(6)
2/3 2.25 0.01 40 93% 0.55102(1)2.4 0.01 40 93% 0.56861(1)2.55 0.01 40 93% 0.58435(1)
0.0 5.8 0.02 20 69% 0.56763(5)5.9 0.02 20 69% 0.58190(3)6.0 0.02 20 68% 0.59364(2)
164 1.0 1.3 0.01 20 82% 0.57840(1)1.42 0.01 20 82% 0.59167(1)
2/3 2.55 0.01 20 88% 0.58428(2)2.7 0.01 20 87% 0.59862(1)
0.0 6.0 0.01 20 89% 0.59382(5)6.13 0.01 40 88% 0.60711(4)
204 1.0 1.3 0.01 20 72% 0.57847(9)1.42 0.01 20 74% 0.59165(1)
2/3 2.55 0.01 20 82% 0.58438(2)2.7 0.01 20 82% 0.59865(1)
0.0 6.0 0.015 20 53% 0.59382(4)6.13 0.01 20 83% 0.60716(3)
![Page 22: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/22.jpg)
Negative mass Wilson fermion (quenched)
With s=0.6. Topology conserving gauge action (1/ε=1,2/3,0) Algorithm: HMC + pseudofermion Lattice size : 144,164 . 1 trajectory = 10 - 15 molecular dynamics steps
with stepsize Δτ= 0.01.
3. Lattice simulations
The simulations were done on the Alpha work station at YITP and SX-5 at RCNP.
size 1/ε β Δτ Nmds acceptance
Plaquette
144 1.0 0.75 0.01 15 80% 0.52287(4)2/3 1.8 0.01 15 86% 0.52930(8)0.0 5.0 0.01 15 88% 0.55466(9)
164 1.0 0.8 0.01 8 75% 0.53115(4)2/3 1.75 0.01 10 91% 0.52309(3)0.0 5.2 0.01 7 90% 0.57567(4)
![Page 23: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/23.jpg)
Implementation of the overlap operator
We use the implicit restarted Arnoldi method (ARPACK) to calculate the eigenvalues of .
To compute , we use the Chebyshevpolynomial approximation after subtracting 10 lowest eigenmodes exactly.
Eigenvalues are calculated with ARPACK, too.
3. Lattice simulations
ARPACK, available from http://www.caam.rice.edu/software/
![Page 24: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/24.jpg)
Initial configurationFor topologically non-trivial initial configuration, we use a discretized version of instanton solution on 4D torus;
which gives constant field strength with arbitrary Q.
3. Lattice simulations
A.Gonzalez-Arroyo,hep-th/9807108, M.Hamanaka,H.Kajiura,Phys.Lett.B551,360(‘03)
![Page 25: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/25.jpg)
New cooling method to measure QWe “cool” the configuration smoothly by performing HMC steps with exponentially increasing (The bound is always
satisfied along the cooling). ⇒ We obtain a “cooled ” configuration close to the
classical background at very high β ~ 106, (after 40-50 steps) then
gives a number close to the index of the overlap operator. NOTE: 1/εcool= 2/3 is useful for 1/ε= 0.0 .
3. Lattice simulations
The agreement of Q with cooling and the index ofoverlap D is roughly (with only 20-80 samples)
~ 90-95% for 1/ε= 1.0 and 2/3. ~ 60-70% for 1/ε=0.0 (plaquette action)
![Page 26: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/26.jpg)
The static quark potentialIn the following, we assume Q does not affect the Wilson loops. ( initial Q=0 )
1. We measure the Wilson loops, in6 different spatial direction,
using smearing. G.S.Bali,K.Schilling,Phys.Rev.D47,661(‘93)
2. The potential is extracted as .
3. From results, we calculate the force following ref S.Necco,R.Sommer,Nucl.Phys.B622,328(‘02)
4. Sommer scales are determined by
4. Results quenchedWith det Hw2
(Preliminary)
![Page 27: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/27.jpg)
The static quark potentialIn the following, we assume Q does not affect the Wilson loops. ( initial Q=0 )
1. We measure the Wilson loops, in6 different spatial direction,
using smearing. G.S.Bali,K.Schilling,Phys.Rev.D47,661(‘93)
2. The potential is extracted as .
3. From results, we calculate the force following ref S.Necco,R.Sommer,Nucl.Phys.B622,328(‘02)
4. Sommer scales are determined by
4. Results
![Page 28: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/28.jpg)
The static quark potential
Here we assume r0 ~ 0.5 fm.
4. Results
size 1/ε β samples r0/a rc/a a rc/r0124 1.0 1.0 3800 3.257(30) 1.7081(50) ~0.15fm 0.5244(52)
1.2 3800 4.555(73) 2.319(10) ~0.11fm 0.5091(81)1.3 3800 5.140(50) 2.710(14) ~0.10fm 0.5272(53)
2/3 2.25 3800 3.498(24) 1.8304(60) ~0.14fm 0.5233(41)2.4 3800 4.386(53) 2.254(16) ~0.11fm 0.5141(61)2.55 3800 5.433(72) 2.809(18) ~0.09fm 0.5170(67)
164 1.0 1.3 2300 5.240(96) 2.686(13) ~0.10fm 0.5126(98)1.42 2247 6.240(89) 3.270(26) ~0.08fm 0.5241(83)
2/3 2.55 1950 5.290(69) 2.738(15) ~0.09fm 0.5174(72)2.7 2150 6.559(76) 3.382(22) ~0.08fm 0.5156(65)
Continuum limit (Necco,Sommer ‘02) 0.5133(24)
quenched
![Page 29: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/29.jpg)
The static quark potential
4. Results
size 1/ε β samples r0/a rc/a a rc/r0164 1.0 0.8 26 5.7(1.0) 3.62(41) ~0.09fm 0.64(16)
2/3 1.75 23 6.26(36) 3.400(80) ~0.08fm 0.543(28)0 5.2 80 6.16(19) 3.441(93) ~0.08fm 0.559(22)
144 1.0 0.75 28 4.97(58) 2.578(75) ~0.1fm 0.520(62)2/3 1.8 68 5.68(90) 2.524(92) ~0.09fm 0.445(72)0 5.0 24 6.1(1.2) 3.48(34) ~0.08fm 0.57(10)
Continuum limit (Necco,Sommer ‘02) 0.5133(24)
With det Hw2 (Preliminary)
![Page 30: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/30.jpg)
Renormalization of the couplingThe renormalized coupling in Manton-scheme is defined
where is the tadpole improved bare coupling:
where P is the plaquette expectation value.
4. Results
R.K.Ellis,G.Martinelli, Nucl.Phys.B235,93(‘84)Erratum-ibid.B249,750(‘85)
quenched
![Page 31: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/31.jpg)
The stability of the topological charge
The stability of Q for 4D QCD is proved only when ε < εmax ~ 1/30 ,which is not practical…
Topology preservation should be perfectBut large scaling violations??
4. Results
SG
ε< 1/30
Q=0 ε=∞ Q=1
SG
ε= 1.0
Q=0 Q=1
If the barrier is high enough, Q may be fixed.
![Page 32: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/32.jpg)
The stability of the topological chargeWe measure Q using cooling per 20 trajectories
: auto correlation for the plaquette
: total number of trajectories : (lower bound of ) number of topology changes
We define “stability” by the ratio of topology change rate ( ) over the plaquette autocorrelation( ).
Note that this gives only the upper bound of the stability.
4. Results
M.Luescher, hep-lat/0409106 Appendix E.
![Page 33: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/33.jpg)
size 1/ε β r0/a Trj τplaq #Q Q stability124 1.0 1.0 3.398(55) 18000 2.91(33) 696 9
2/3 2.25 3.555(39) 18000 5.35(79) 673 50.0 5.8 [3.668(12)] 18205 30.2(6.6) 728 11.0 1.2 4.464(65) 18000 1.59(15) 265 432/3 2.4 4.390(99) 18000 2.62(23) 400 170.0 5.9 [4.483(17)] 27116 13.2(1.5) 761 31.0 1.3 5.240(96) 18000 1.091(70) 69 2392/3 2.55 5.290(69) 18000 2.86(33) 123 510.0 6.0 [5.368(22)] 27188 15.7(3.0) 304 6
164 1.0 1.3 5.240(96) 11600 3.2(6) 78 462/3 2.55 5.290(69) 12000 6.4(5) 107 180.0 6.0 [5.368(22)] 3500 11.7(3.9) 166 1.81.0 1.42 6.240(89) 5000 2.6(4) 2 9612/3 2.7 6.559(76) 14000 3.1(3) 6 7520.0 6.13 [6.642(-)] 5500 12.4(3.3) 22 20
204 1.0 1.3 5.240(96) 1240 2.6(5) 14 342/3 2.55 5.290(69) 1240 3.4(7) 15 240.0 6.0 [5.368(22)] 1600 14.4(7.8) 37 31.0 1.42 6.240(89) 7000 3.8(8) 29 632/3 2.7 6.559(76) 7800 3.5(6) 20 1100.0 6.13 [6.642(-)] 1298 9.3(2.8) 4 35
quenched
![Page 34: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/34.jpg)
size 1/ε β r0/a Trj τplaq #Q Q stability164 1.0 0.8 5.7(1.0) 520 12(5) 0 >43
2/3 1.75 6.26(36) 460 10(4) 0 >460.0 5.2 6.16(19) 1614 51(31) 0 >32
144 1.0 0.75 4.97(58) 560 5(2) 0 >1122/3 1.8 5.68(90) 1360 14(5) 0 >970.0 5.0 6.1(1.2) 480 11(5) 0 >44
4. ResultsWith det Hw2 (Preliminary)
Topology conservation seems perfect !
![Page 35: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/35.jpg)
The overlap Dirac operatorWe expect Low-modes of Hw are suppressed.
⇒ the Chebyshev approximation is improved.
: The condition number: order of polynomial : constants independent of V, β, ε…
Locality is improved.
4. Results
![Page 36: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/36.jpg)
The condition number
The gain is about a factor 2-3.
4. Results
size 1/ε β r0/a Q stability 1/κ P(<0.1)204 1.0 1.3 5.240(96) 34 0.0148(14) 0.090(14)
2/3 2.55 5.290(69) 24 0.0101(08) 0.145(12)0.0 6.0 5.368(22) 3 0.0059(34) 0.414(29)1.0 1.42 6.240(89) 63 0.0282(21) 0.031(10)2/3 2.7 6.559(76) 110 0.0251(19) 0.019(18)0.0 6.13 6.642(-) 35 0.0126(15) 0.084(14)
164 1.0 1.42 6.240(89) 961 0.0367(21) 0.007(5)2/3 2.7 6.559(76) 752 0.0320(19) 0.020(8)0.0 6.13 6.642(-) 20 0.0232(17) 0.030(10)
quenched
![Page 37: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/37.jpg)
The condition number
4. Results
size 1/ε β r0/a Q stability hwmin P(<0.1)
164 1.0 0.8 5.7(1.0) >43 0.1823(33) 02/3 1.75 6.26(36) >46 0.1284(13) 0.080.0 5.2 6.16(19) >32 0.2325(17) 0.05
quenched 0 6.13 6.642 20 0.139(10) 0.03
With det Hw2 (Preliminary)
![Page 38: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/38.jpg)
The localityFor
should exponentially decay.1/a~0.08fm (with 4 samples),no remarkable improvement of locality is seen…
⇒ lower beta?
+ : beta = 1.42, 1/e=1.0
× : beta = 2.7, 1/e=2/3
* : beta = 6.13, 1/e=0.0
quenched
4. Results
![Page 39: Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto.](https://reader030.fdocuments.net/reader030/viewer/2022012922/56649f125503460f94c25dc7/html5/thumbnails/39.jpg)
We find New cooling method does work. In quenched study, the lattice spacing can be determined in a co
nventional manner, ant the quark potential show no large deviation from the continuum limit. For det Hw2, we need more configurations.
Q can be fixed. . No clear improvement of the locality (for high beta). The numerical cost of Chebyshev approximation would be 1.2-2.
5 times better than that with plaquette action.
5. Conclusion and Outlook
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For future works, we would like to try
Including twisted mass ghost,
Summation of different topology Dynamical overlap fermion at fixed topology
5. Conclusion and Outlook
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Topology dependence
Q dependence of the quark potential seems week
as we expected.
4. Results
size 1/ε β Initial Q Q stability plaquette r0/a rc/r0164 1.0 1.42 0 961 0.59165(1) 6.240(89) 0.5126(98)
1.42 -3 514 0.59162(1) 6.11(13) 0.513(12)