Topology conserving gauge actionTopology conserving gauge actionand and

the overlap Dirac operatorthe overlap Dirac operator

Hidenori FukayaHidenori FukayaYukawa Institute, Kyoto Univ. Yukawa Institute, Kyoto Univ.

Collaboration withCollaboration with S.Hashimoto (KEK,Sokendai), T.Hirohashi (Kyoto Univ.),S.Hashimoto (KEK,Sokendai), T.Hirohashi (Kyoto Univ.), H.Matsufuru(KEK), K.Ogawa(Sokendai), T.Onogi(YITP)H.Matsufuru(KEK), K.Ogawa(Sokendai), T.Onogi(YITP)

ContentsContents

1.1. IntroductionIntroduction2.2. Lattice simulationsLattice simulations3.3. The static quark potentialThe static quark potential4.4. Stability of the topological chargeStability of the topological charge5.5. The overlap Dirac operatorThe overlap Dirac operator6.6. Conclusion and outlookConclusion and outlook

1.1. IntroductionIntroduction

Lattice regularization of the gauge theory is a verLattice regularization of the gauge theory is a very powerful tool to analyze strong coupling regime y powerful tool to analyze strong coupling regime but it spoils a lot of symmetries…but it spoils a lot of symmetries…

Translational symmetryTranslational symmetryLorentz invarianceLorentz invarianceChiral symmetry or topologyChiral symmetry or topologySupersymmetry…Supersymmetry… 　　

Nielsen-Ninomiya theoremNielsen-Ninomiya theorem

Any local Dirac operator satisfying chiral symmetry;Any local Dirac operator satisfying chiral symmetry; has unphysical poles (doublerhas unphysical poles (doubler

s).s).

Example – free naïve fermion –Example – free naïve fermion – ContinuumContinuum has no doubler.has no doubler. LatticeLattice

has unphysical poles at has unphysical poles at . . Wilson Dirac operatorWilson Dirac operator

Doublers are decoupled (heavy) but no chiral symmetry.Doublers are decoupled (heavy) but no chiral symmetry.

Nucl.Phys.B185,20 Nucl.Phys.B185,20 (‘81),Nucl.Phys.B193,173 (‘81)(‘81),Nucl.Phys.B193,173 (‘81)

Ginsparg-Wilson relationGinsparg-Wilson relationThe Neuberger’s overlap operator:The Neuberger’s overlap operator:

satisfying the Ginsparg-Wilson relation:satisfying the Ginsparg-Wilson relation:

realizes ‘modified’ exact chiral symmetry on the lattice;realizes ‘modified’ exact chiral symmetry on the lattice;the action is invariant underthe action is invariant under

NOTENOTE Expansion in Wilson Dirac operator ⇒Expansion in Wilson Dirac operator ⇒ 　　 No doubler.No doubler. Fermion measure is not invariant;Fermion measure is not invariant;

⇒⇒ 　　 chiral anomaly, index theoremchiral anomaly, index theorem

locality? or smoothness?locality? or smoothness?

Phys.Rev.D25,2649(‘Phys.Rev.D25,2649(‘82)82)

Phys.Lett.B417,141(‘Phys.Lett.B417,141(‘98)98)

The locality of the overlap operatorThe locality of the overlap operator

If the gauge fields are smooth;If the gauge fields are smooth;

with a (small) fixed number ε , it gives a lower bowith a (small) fixed number ε , it gives a lower bound tound to

⇒ ⇒ (exponential) locality and smoothness of the Di(exponential) locality and smoothness of the Dirac operator;rac operator;

⇒ ⇒ Stability of the index (topological charge);Stability of the index (topological charge);

P.Hernandez,K.Jansen,M.Luescher,Nucl.Phys.B552,363P.Hernandez,K.Jansen,M.Luescher,Nucl.Phys.B552,363(‘99)(‘99)

Topology conserving gauge actionTopology conserving gauge action

The bound The bound is is

automatically satisfied with the actionautomatically satisfied with the action

Hybrid Monte Carlo (HMC) algorithmHybrid Monte Carlo (HMC) algorithm Heat bath algorithm = local and large updatesHeat bath algorithm = local and large updates HMC algorithm = global and HMC algorithm = global and small small updatesupdates

⇒⇒ 　　 Q may be fixed along the simulationsQ may be fixed along the simulations

M.Luescher,Phys.Lett.B428,342(‘98),Nucl.Phys.B549,2M.Luescher,Phys.Lett.B428,342(‘98),Nucl.Phys.B549,295(‘99)95(‘99)

- Motivations -- Motivations -

Efficient sampling of the configurations with fixed Q. Efficient sampling of the configurations with fixed Q. ⇒ ε- regime with fixed Q⇒ ε- regime with fixed Q

Improvement of the locality of the overlap operator.Improvement of the locality of the overlap operator. Lower numerical costs of the overlap operator Lower numerical costs of the overlap operator

⇒⇒ 　　 dynamical overlap fermionsdynamical overlap fermions Anyway, it’s interesting… Anyway, it’s interesting…

We studied the We studied the Nf=2 massive Schwinger modelNf=2 massive Schwinger model

with a theta termwith a theta term using using the topology conserving acthe topology conserving actiontion and and the domain-wall fermion action;the domain-wall fermion action;

HF,T.Onogi,Phys.Rev.D68,074503(‘03),Phys.Rev.D70,054508(’04)HF,T.Onogi,Phys.Rev.D68,074503(‘03),Phys.Rev.D70,054508(’04)

Good chiral behavior (consistent with theGood chiral behavior (consistent with the

bosonization approach).bosonization approach). Perfect stability of Q with ε= 1.0.Perfect stability of Q with ε= 1.0.

- Our goals -- Our goals -

Scaling studies (to show validity in Scaling studies (to show validity in 4D quenched QCD4D quenched QCD.).) determination of the lattice spacingdetermination of the lattice spacing scaling violations of the quark potentialscaling violations of the quark potential

How stable How stable QQ ? ? Locality and numerical cost of GW fermion improved ? Locality and numerical cost of GW fermion improved ?

c.f. S.Shcheredin, W.Bietenholz, K.Jansen, K.I.Nagai, S.Necco and L.Scorzato, hep-lat/0409073 , hep-lat/0412017 .

2.2. Lattice simulationsLattice simulations Action: Luescher action (Action: Luescher action (quenchedquenched))

with 1/ε= 1.0, 2/3, 0.0 (=plaquette action) .with 1/ε= 1.0, 2/3, 0.0 (=plaquette action) . Algorithm: The standard HMC method.Algorithm: The standard HMC method. Lattice size : 12Lattice size : 1244,16,1644,20,2044 . . 1 trajectory = 20 - 40 molecular dynamics steps with 1 trajectory = 20 - 40 molecular dynamics steps with

stepsize Δτ= 0.01 - 0.02.stepsize Δτ= 0.01 - 0.02.

The simulations were done on the Alpha work station at YITP and SX-5 at RCNP.

sizesize 1/ε1/ε ββ ΔτΔτ NNmdsmds acceptancacceptancee

PlaquettePlaquette

121244 1.01.0 1.01.0 0.010.01 4040 89%89% 0.539127(9)0.539127(9)1.21.2 0.010.01 4040 90%90% 0.566429(6)0.566429(6)1.31.3 0.010.01 4040 90%90% 0.578405(6)0.578405(6)

2/32/3 2.252.25 0.010.01 4040 93%93% 0.55102(1)0.55102(1)2.42.4 0.010.01 4040 93%93% 0.56861(1)0.56861(1)

2.552.55 0.010.01 4040 93%93% 0.58435(1)0.58435(1)0.00.0 5.85.8 0.020.02 2020 69%69% 0.56763(5)0.56763(5)

5.95.9 0.020.02 2020 69%69% 0.58190(3)0.58190(3)6.06.0 0.020.02 2020 68%68% 0.59364(2)0.59364(2)

161644 1.01.0 1.31.3 0.010.01 2020 82%82% 0.57840(1)0.57840(1)1.421.42 0.010.01 2020 82%82% 0.59167(1)0.59167(1)

2/32/3 2.552.55 0.010.01 2020 88%88% 0.58428(2)0.58428(2)2.72.7 0.010.01 2020 87%87% 0.59862(1)0.59862(1)

0.00.0 6.06.0 0.010.01 2020 89%89% 0.59382(5)0.59382(5)6.136.13 0.010.01 4040 88%88% 0.60711(4)0.60711(4)

202044 1.01.0 1.31.3 0.010.01 2020 72%72% 0.57847(9)0.57847(9)1.421.42 0.010.01 2020 74%74% 0.59165(1)0.59165(1)

2/32/3 2.552.55 0.010.01 2020 82%82% 0.58438(2)0.58438(2)2.72.7 0.010.01 2020 82%82% 0.59865(1)0.59865(1)

0.00.0 6.06.0 0.0150.015 2020 53%53% 0.59382(4)0.59382(4)6.136.13 0.010.01 2020 83%83% 0.60716(3)0.60716(3)

The overlap Dirac operatorThe overlap Dirac operator

We use the implicit restarted Arnoldi method We use the implicit restarted Arnoldi method (ARPACK) to calculate the eigenvalues of .(ARPACK) to calculate the eigenvalues of .

To compute , we use the ChebyshevTo compute , we use the Chebyshevpolynomial approximation after subtracting 10 lowest polynomial approximation after subtracting 10 lowest eigenmodes exactly.eigenmodes exactly.

We set s=0.6We set s=0.6

ARPACK, available from http://www.caam.rice.edu/software/ARPACK, available from http://www.caam.rice.edu/software/

Initial configurationInitial configurationFor topologically non-trivial initial configuration, we use For topologically non-trivial initial configuration, we use a discretized version of a discretized version of instanton solutioninstanton solution on 4D torus; on 4D torus;

which gives which gives constant field strengthconstant field strength with arbitrary with arbitrary QQ..

A.Gonzalez-Arroyo,hep-th/9807108, A.Gonzalez-Arroyo,hep-th/9807108, M.Hamanaka,H.Kajiura,Phys.Lett.B551,36M.Hamanaka,H.Kajiura,Phys.Lett.B551,360(‘03)0(‘03)

NewNew cooling method to measure Q cooling method to measure Q

We “We “coolcool” the configuration smoothly by ” the configuration smoothly by performing HMC steps with exponentially performing HMC steps with exponentially increasing increasing

(The bound (The bound is always is always satisfiedsatisfied along the cooling). along the cooling).

⇒ ⇒ We obtain a “We obtain a “cooledcooled ” configuration close to the ” configuration close to the classical background at very high βclassical background at very high β ～～ 101066, (after , (after 40-50 steps) then40-50 steps) then

gives a number close to the index of the overlap gives a number close to the index of the overlap operator.operator.

NOTE: 1/εNOTE: 1/εcoolcool= 2/3 is useful for 1/ε= 0.0 .= 2/3 is useful for 1/ε= 0.0 .

CoolingCooling -Example: 2D QED- -Example: 2D QED- - instanton anti-instanton pair on 16- instanton anti-instanton pair on 1622 lattice - lattice -

Eigen values of γ ５ D

Eigen function （ chirality + ）Eigen function （ chirality －）

Let’s cool it !Let’s cool it !

CoolingCooling -Example: 2D QED- -Example: 2D QED- - 2 instanton and 1 anti-instanton -- 2 instanton and 1 anti-instanton -

Eigenvalues

→1 instanton

Let’s cool it !Let’s cool it !

CoolingCooling -4D QCD results- -4D QCD results-

The agreement of The agreement of Q with coolingQ with cooling and and the index ofthe index ofoverlap Doverlap D is roughly (with only 20-80 samples) is roughly (with only 20-80 samples) ~ 90-95% for 1/ε= 1.0 and 2/3.~ 90-95% for 1/ε= 1.0 and 2/3. ~ 60-70% for 1/ε=0.0 (plaquette action)~ 60-70% for 1/ε=0.0 (plaquette action)

3.3. The static quark The static quark potentialpotential In the following, we assume Q does not affect the In the following, we assume Q does not affect the

Wilson loops. ( initial Q=0 )Wilson loops. ( initial Q=0 )

We measure the Wilson loops,We measure the Wilson loops, inin6 different spatial direction,6 different spatial direction,

using smearing. using smearing. G.S.Bali,K.Schilling,Phys.Rev.D47,661(‘93)G.S.Bali,K.Schilling,Phys.Rev.D47,661(‘93)

The potential is extracted as The potential is extracted as .. From From results, we calculate the force following results, we calculate the force following S.S.

Necco,R.Sommer,Nucl.Phys.B622,328(‘02)Necco,R.Sommer,Nucl.Phys.B622,328(‘02) Sommer scales are determined bySommer scales are determined by

Assuming rAssuming r00 ~ 0.5 fm, we obtain ~ 0.5 fm, we obtain

(preliminary) (preliminary)

sizesize 1/ε1/ε ββ samplessamples r0/ar0/a rc/arc/a aa rc/r0rc/r0121244 1.01.0 1.01.0 38003800 3.257(303.257(30

))1.7081(50)1.7081(50) ~0.15fm~0.15fm 0.5244(520.5244(52

))1.21.2 38003800 4.555(734.555(73

))2.319(10)2.319(10) ~0.11fm~0.11fm 0.5091(810.5091(81

))1.31.3 38003800 5.140(505.140(50

))2.710(14)2.710(14) ~0.10fm~0.10fm 0.5272(530.5272(53

))2/32/3 2.252.25 38003800 3.498(243.498(24

))1.8304(60)1.8304(60) ~0.14fm~0.14fm 0.5233(410.5233(41

))2.42.4 38003800 4.386(534.386(53

))2.254(16)2.254(16) ~0.11fm~0.11fm 0.5141(610.5141(61

))2.552.55 38003800 5.433(725.433(72

))2.809(18)2.809(18) ~0.09fm~0.09fm 0.5170(670.5170(67

))161644 1.01.0 1.31.3 23002300 5.240(965.240(96

))2.686(13)2.686(13) ~0.10fm~0.10fm 0.5126(980.5126(98

))1.421.42 22472247 6.240(896.240(89

))3.270(26)3.270(26) ~0.08fm~0.08fm 0.5241(830.5241(83

))2/32/3 2.552.55 19501950 5.290(695.290(69

))2.738(15)2.738(15) ~0.09fm~0.09fm 0.5174(720.5174(72

))2.72.7 21502150 6.559(766.559(76

))3.382(22)3.382(22) ~0.08fm~0.08fm 0.5156(650.5156(65

))Continuum limit (Necco,Sommer ‘02)Continuum limit (Necco,Sommer ‘02) 0.5133(240.5133(24

))

The quark potential itself is good.The quark potential itself is good.

Perturbative renormalization of the couplingPerturbative renormalization of the coupling

At 1-loop level, renormalized At 1-loop level, renormalized couplings have similar values couplings have similar values but the convergence of the pbut the convergence of the perturbative series is much werturbative series is much worse for 1/ε= 1, 2/3. orse for 1/ε= 1, 2/3. Tadpole improvement at 2-loTadpole improvement at 2-loop level may be required or nop level may be required or non-perturbative approach shoon-perturbative approach should be done.uld be done.

R.K.Ellis,G.Martinelli, R.K.Ellis,G.Martinelli, Nucl.Phys.B235,93(‘84)Nucl.Phys.B235,93(‘84)Erratum-ibid.B249,750(‘85)Erratum-ibid.B249,750(‘85)

The stability of The stability of QQ for 4D QCD is proved only when for 4D QCD is proved only when εε ＜ ＜ εεmaxmax ～～ 1/301/30 ,which is not practical… ,which is not practical…

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.

4.4. Stability of the topological Stability of the topological chargecharge

Let us search the parameters;Let us search the parameters;

β,β, 　　 ε,ε, 　　 V which can fix Q.V which can fix Q.

We measure Q using We measure Q using coolingcooling per 20 trajectories per 20 trajectories

: auto correlation for the plaquette: auto correlation for the plaquette

: total number of trajectories: total number of trajectories : (lower bound of ) number of topology changes: (lower bound of ) number of topology changes

We define “stability” by the ratio of topology changeWe define “stability” by the ratio of topology change rate ( ) over the plaquette autocorrelation( ).rate ( ) over the plaquette autocorrelation( ).

Note that this gives only the upper bound of the stability, Note that this gives only the upper bound of the stability, since there may be topology changes during every 20 since there may be topology changes during every 20 trajectories.trajectories.

M.Luescher, hep-lat/0409106 Appendix E.M.Luescher, hep-lat/0409106 Appendix E.

sizesize 1/ε1/ε ββ r0/ar0/a TrjTrj τplaqτplaq #Q#Q Q stabilityQ stability121244 1.01.0 1.01.0 3.398(55)3.398(55) 1800018000 2.91(33)2.91(33) 6969

6699

2/32/3 2.252.25 3.555(39)3.555(39) 1800018000 5.35(79)5.35(79) 676733

55

0.00.0 5.85.8 [3.668(12)][3.668(12)] 1820518205 30.2(6.6)30.2(6.6) 727288

11

1.01.0 1.21.2 4.464(65)4.464(65) 1800018000 1.59(15)1.59(15) 262655

4343

2/32/3 2.42.4 4.390(99)4.390(99) 1800018000 2.62(23)2.62(23) 404000

1717

0.00.0 5.95.9 [4.483(17)][4.483(17)] 2711627116 13.2(1.5)13.2(1.5) 767611

33

1.01.0 1.31.3 5.240(96)5.240(96) 1800018000 1.091(70)1.091(70) 6969 2392392/32/3 2.552.55 5.290(69)5.290(69) 1800018000 2.86(33)2.86(33) 1212

335151

0.00.0 6.06.0 [5.368(22)][5.368(22)] 2718827188 15.7(3.0)15.7(3.0) 303044

66

161644 1.01.0 1.31.3 5.240(96)5.240(96) 1160011600 3.2(6)3.2(6) 7878 46462/32/3 2.552.55 5.290(69)5.290(69) 1200012000 6.4(5)6.4(5) 1010

771818

0.00.0 6.06.0 [5.368(22)][5.368(22)] 35003500 11.7(3.9)11.7(3.9) 161666

1.81.8

1.01.0 1.421.42 6.240(89)6.240(89) 50005000 2.6(4)2.6(4) 22 9619612/32/3 2.72.7 6.559(76)6.559(76) 1400014000 3.1(3)3.1(3) 66 7527520.00.0 6.136.13 [6.642(-)][6.642(-)] 55005500 12.4(3.3)12.4(3.3) 2222 2020

202044 1.01.0 1.31.3 5.240(96)5.240(96) 12401240 2.6(5)2.6(5) 1414 34342/32/3 2.552.55 5.290(69)5.290(69) 12401240 3.4(7)3.4(7) 1515 24240.00.0 6.06.0 [5.368(22)][5.368(22)] 16001600 14.4(7.8)14.4(7.8) 3737 331.01.0 1.421.42 6.240(89)6.240(89) 70007000 3.8(8)3.8(8) 2929 63632/32/3 2.72.7 6.559(76)6.559(76) 78007800 3.5(6)3.5(6) 2020 1101100.00.0 6.136.13 [6.642(-)][6.642(-)] 12981298 9.3(2.8)9.3(2.8) 44 3535

• Stability is better for smaller ε, smaller L , smaller a.Stability is better for smaller ε, smaller L , smaller a.• A significantly good stability can be A significantly good stability can be

seen seen forfor

⇒⇒ 　　 maybe valid in maybe valid in ε- regimeε- regime..

Q dependence of the quark potentialQ dependence of the quark potential

seems week as we expected.seems week as we expected.

sizesize 1/ε1/ε ββ Initial QInitial Q Q Q stabilitystability

plaquetteplaquette r0/ar0/a rc/r0rc/r0

161644 1.01.0 1.41.422

00 961961 0.59165(1)0.59165(1) 6.240(89)6.240(89) 0.5126(980.5126(98))

1.41.422

-3-3 514514 0.59162(1)0.59162(1) 6.11(13)6.11(13) 0.513(12)0.513(12)

It is known that the lowest eigen value of It is known that the lowest eigen value of is related is related toto

LocalityLocality is guaranteed if is guaranteed if never touches zero. never touches zero. The order of Chebyshev polynomialThe order of Chebyshev polynomial can be reduced; can be reduced;

: the condition number: the condition number

: a random vector: a random vector

5.5. The overlap Dirac operatorThe overlap Dirac operator

The localityThe localityFor For

should exponentially should exponentially decay.decay.

1/a~0.08fm 1/a~0.08fm (with 4 samples),(with 4 samples),no remarkable no remarkable improvement of improvement of locality is seen…locality is seen…

⇒⇒ 　　 lower beta?lower beta?

＋ : beta = 1.42, 1/e=1.0

× : beta = 2.7, 1/e=2/3

＊ : beta = 6.13, 1/e=0.0

The condition numberThe condition number and Chebyshev aand Chebyshev approximationpproximation

sizesize 1/ε1/ε ββ r0/ar0/a Q stabilityQ stability202044 1.01.0 1.31.3 5.240(96)5.240(96) 3434 0.0148(14)0.0148(14) 0.03970(20.03970(2

9)9)2/32/3 2.552.55 5.290(69)5.290(69) 2424 0.0101(08)0.0101(08) 0.03651(20.03651(2

7)7)0.00.0 6.06.0 5.368(22)5.368(22) 33 0.0059(34)0.0059(34) 0.02766(40.02766(4

6)6)1.01.0 1.421.42 6.240(89)6.240(89) 6363 0.0282(21)0.0282(21) 0.04765(30.04765(3

2)2)2/32/3 2.72.7 6.559(76)6.559(76) 110110 0.0251(19)0.0251(19) 0.04646(30.04646(3

7)7)0.00.0 6.136.13 6.642(-)6.642(-) 3535 0.0126(15)0.0126(15) 0.03775(50.03775(5

0)0)161644 1.01.0 1.421.42 6.240(89)6.240(89) 961961 0.0367(21)0.0367(21) 0.05233(20.05233(2

6)6)2/32/3 2.72.7 6.559(76)6.559(76) 752752 0.0320(19)0.0320(19) 0.05117(20.05117(2

9)9)0.00.0 6.136.13 6.642(-)6.642(-) 2020 0.0232(17)0.0232(17) 0.04384(30.04384(3

8)8)

We use 10 different Npol and 4 different cWe use 10 different Npol and 4 different confs for each parameter set.onfs for each parameter set. For 1/ε=1.0,For 1/ε=1.0, is is 1.6~2.51.6~2.5 times larger times larger

(( is is 1.2~1.41.2~1.4 times larger) than that with the times larger) than that with the plaquette action.plaquette action.

and are nor independent.and are nor independent.

Topology conserving gauge action may be practical.Topology conserving gauge action may be practical.

New New coolingcooling method does work. method does work. The lattice spacing can be determined in a conventional mThe lattice spacing can be determined in a conventional m

anner, ant the quark potential show no large deviation from anner, ant the quark potential show no large deviation from the continuum limit.the continuum limit.

Q can be stable forQ can be stable for .. No improvement of the locality (for high beta).No improvement of the locality (for high beta). The numerical cost of Chebyshev approximation would be The numerical cost of Chebyshev approximation would be

1.2-2.5 times better1.2-2.5 times better than that with plaquette action. than that with plaquette action.

6.6. Conclusion and outlookConclusion and outlook

OutlooksOutlooks

Accept/reject for topology changes to fix Q complAccept/reject for topology changes to fix Q completely (per every 100trj would be enough.). etely (per every 100trj would be enough.).

Quenched ε-regime with fixed Q.Quenched ε-regime with fixed Q. Nf=2 full QCD in the ε-regime with the overlap ferNf=2 full QCD in the ε-regime with the overlap fer

mion.mion.