Topology conserving gauge action and the overlap Dirac operator Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto (KEK,Sokendai), T.Hirohashi (Kyoto Univ.), H.Matsufuru(KEK), K.Ogawa(Sokendai), T.Onogi(YITP)

Contents Introduction Lattice simulations The static quark potential Stability of the topological charge The overlap Dirac operator Conclusion and outlook

1. Introduction 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…　

Nielsen-Ninomiya theorem Any local Dirac operator satisfying chiral symmetry; has unphysical poles (doublers). Example – free naïve fermion – Continuum has no doubler. Lattice has unphysical poles at . Wilson Dirac operator Doublers are decoupled (heavy) but no chiral symmetry. Nucl.Phys.B185,20 (‘81),Nucl.Phys.B193,173 (‘81)

locality? or smoothness? 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 locality? or smoothness? Phys.Lett.B417,141(‘98) Phys.Rev.D25,2649(‘82)

The locality of the overlap operator If the gauge fields are smooth; with a (small) fixed number ε , it gives a lower bound to ⇒ (exponential) locality and smoothness of the Dirac operator; ⇒ Stability of the index (topological charge); P.Hernandez,K.Jansen,M.Luescher,Nucl.Phys.B552,363(‘99)

Topology conserving gauge action The bound is automatically satisfied with the action Hybrid Monte Carlo (HMC) algorithm Heat bath algorithm = local and large updates HMC algorithm = global and small updates ⇒　Q may be fixed along the simulations M.Luescher,Phys.Lett.B428,342(‘98),Nucl.Phys.B549,295(‘99)

- Motivations - We studied the Nf=2 massive Schwinger model with a theta term using the topology conserving action and the domain-wall fermion action; HF,T.Onogi,Phys.Rev.D68,074503(‘03),Phys.Rev.D70,054508(’04) Good chiral behavior (consistent with the bosonization approach). Perfect stability of Q with ε= 1.0. Efficient sampling of the configurations with fixed Q. ⇒ ε- regime with fixed Q Improvement of the locality of the overlap operator. Lower numerical costs of the overlap operator ⇒　dynamical overlap fermions Anyway, it’s interesting…

- Our goals - Scaling studies (to show validity in 4D quenched QCD.) determination of the lattice spacing scaling violations of the quark potential How stable Q ? 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. Lattice simulations Action: Luescher action (quenched) size 1/ε β Δτ Nmds acceptance Plaquette 124 1.0 0.01 40 89% 0.539127(9) 1.2 90% 0.566429(6) 1.3 0.578405(6) 2/3 2.25 93% 0.55102(1) 2.4 0.56861(1) 2.55 0.58435(1) 0.0 5.8 0.02 20 69% 0.56763(5) 5.9 0.58190(3) 6.0 68% 0.59364(2) 164 82% 0.57840(1) 1.42 0.59167(1) 88% 0.58428(2) 2.7 87% 0.59862(1) 0.59382(5) 6.13 0.60711(4) 204 72% 0.57847(9) 74% 0.59165(1) 0.58438(2) 0.59865(1) 0.015 53% 0.59382(4) 83% 0.60716(3) 2. Lattice simulations Action: Luescher 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. The simulations were done on the Alpha work station at YITP and SX-5 at RCNP.

The overlap Dirac operator We use the implicit restarted Arnoldi method (ARPACK) to calculate the eigenvalues of . To compute , we use the Chebyshev polynomial approximation after subtracting 10 lowest eigenmodes exactly. We set s=0.6 ARPACK, available from http://www.caam.rice.edu/software/

For topologically non-trivial initial configuration, we use a discretized version of instanton solution on 4D torus; which gives constant field strength with arbitrary Q. A.Gonzalez-Arroyo,hep-th/9807108, M.Hamanaka,H.Kajiura,Phys.Lett.B551,360(‘03)

New cooling method to measure Q We “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 .

Cooling -Example: 2D QED- - instanton anti-instanton pair on 162 lattice - Eigen function （chirality －） Eigen function （chirality +） Eigen values of γ５D Let’s cool it !

Cooling -Example: 2D QED- - 2 instanton and 1 anti-instanton - →1 instanton Eigenvalues Let’s cool it !

Cooling -4D QCD results- The agreement of Q with cooling and the index of overlap 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)

3. The static quark potential In the following, we assume Q does not affect the Wilson loops. ( initial Q=0 ) We measure the Wilson loops, in 6 different spatial direction, using smearing. G.S.Bali,K.Schilling,Phys.Rev.D47,661(‘93) The potential is extracted as . From results, we calculate the force following S.Necco,R.Sommer,Nucl.Phys.B622,328(‘02) Sommer scales are determined by

Continuum limit (Necco,Sommer ‘02) Assuming r0 ~ 0.5 fm, we obtain (preliminary) size 1/ε β samples r0/a rc/a a rc/r0 124 1.0 3800 3.257(30) 1.7081(50) ~0.15fm 0.5244(52) 1.2 4.555(73) 2.319(10) ~0.11fm 0.5091(81) 1.3 5.140(50) 2.710(14) ~0.10fm 0.5272(53) 2/3 2.25 3.498(24) 1.8304(60) ~0.14fm 0.5233(41) 2.4 4.386(53) 2.254(16) 0.5141(61) 2.55 5.433(72) 2.809(18) ~0.09fm 0.5170(67) 164 2300 5.240(96) 2.686(13) 0.5126(98) 1.42 2247 6.240(89) 3.270(26) ~0.08fm 0.5241(83) 1950 5.290(69) 2.738(15) 0.5174(72) 2.7 2150 6.559(76) 3.382(22) 0.5156(65) Continuum limit (Necco,Sommer ‘02) 0.5133(24)

The quark potential itself is good.

Perturbative renormalization of the coupling At 1-loop level, renormalized couplings have similar values but the convergence of the perturbative series is much worse for 1/ε= 1, 2/3. Tadpole improvement at 2-loop level may be required or non-perturbative approach should be done. R.K.Ellis,G.Martinelli, Nucl.Phys.B235,93(‘84) Erratum-ibid.B249,750(‘85)

4. Stability of the topological charge SG ε= 1.0 Q=0 　　　　 　 Q=1 If the barrier is high enough, Q may be fixed. SG ε< 1/30 Q=0 　ε=∞ 　 Q=1 Let us search the parameters; β,　ε,　V which can fix Q. The stability of Q for 4D QCD is proved only when ε＜ εmax ～1/30 ,which is not practical…

We define “stability” by the ratio of topology change We 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, since there may be topology changes during every 20 trajectories. M.Luescher, hep-lat/0409106 Appendix E.

Stability is better for smaller ε, smaller L , smaller a. size 1/ε β r0/a Trj τplaq #Q Q stability 124 1.0 3.398(55) 18000 2.91(33) 696 9 2/3 2.25 3.555(39) 5.35(79) 673 5 0.0 5.8 [3.668(12)] 18205 30.2(6.6) 728 1 1.2 4.464(65) 1.59(15) 265 43 2.4 4.390(99) 2.62(23) 400 17 5.9 [4.483(17)] 27116 13.2(1.5) 761 3 1.3 5.240(96) 1.091(70) 69 239 2.55 5.290(69) 2.86(33) 123 51 6.0 [5.368(22)] 27188 15.7(3.0) 304 6 164 11600 3.2(6) 78 46 12000 6.4(5) 107 18 3500 11.7(3.9) 166 1.8 1.42 6.240(89) 5000 2.6(4) 2 961 2.7 6.559(76) 14000 3.1(3) 752 6.13 [6.642(-)] 5500 12.4(3.3) 22 20 204 1240 2.6(5) 14 34 3.4(7) 15 24 1600 14.4(7.8) 37 7000 3.8(8) 29 63 7800 3.5(6) 110 1298 9.3(2.8) 4 35 Stability is better for smaller ε, smaller L , smaller a. A significantly good stability can be seen for ⇒　maybe valid in ε- regime.

Q dependence of the quark potential seems week as we expected. size 1/ε β Initial Q Q stability plaquette r0/a rc/r0 164 1.0 1.42 961 0.59165(1) 6.240(89) 0.5126(98) -3 514 0.59162(1) 6.11(13) 0.513(12)

5. The overlap Dirac operator It is known that the lowest eigen value of is related to Locality is guaranteed if never touches zero. The order of Chebyshev polynomial can be reduced; : the condition number : a random vector

should exponentially decay. The locality For 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

We use 10 different Npol and 4 different confs for each parameter set. The condition number and Chebyshev approximation We use 10 different Npol and 4 different confs for each parameter set. size 1/ε β r0/a Q stability 204 1.0 1.3 5.240(96) 34 0.0148(14) 0.03970(29) 2/3 2.55 5.290(69) 24 0.0101(08) 0.03651(27) 0.0 6.0 5.368(22) 3 0.0059(34) 0.02766(46) 1.42 6.240(89) 63 0.0282(21) 0.04765(32) 2.7 6.559(76) 110 0.0251(19) 0.04646(37) 6.13 6.642(-) 35 0.0126(15) 0.03775(50) 164 961 0.0367(21) 0.05233(26) 752 0.0320(19) 0.05117(29) 20 0.0232(17) 0.04384(38) and are nor independent. For 1/ε=1.0, is 1.6~2.5 times larger ( is 1.2~1.4 times larger) than that with the plaquette action.

6. Conclusion and outlook Topology conserving gauge action may be practical. New cooling method does work. The lattice spacing can be determined in a conventional manner, ant the quark potential show no large deviation from the continuum limit. Q can be stable for . No 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.

Outlooks Accept/reject for topology changes to fix Q completely (per every 100trj would be enough.). Quenched ε-regime with fixed Q. Nf=2 full QCD in the ε-regime with the overlap fermion.