Lattice QCD in a fixed topological sector [hep-lat/0603008] Hidenori Fukaya Theoretical Physics laboratory, RIKEN PhD thesis based on Phys.Rev.D73, 014503 (2005)[hep-lat/0510116] Collaboration with S.Hashimoto (KEK,Sokendai), T.Hirohashi (Kyoto Univ.), H.Matsufuru(KEK), K.Ogawa(Nathinal Taiwan Univ.) and T.Onogi(YITP)

Contents Introduction The chiral symmetry and topology Lattice simulations Results Summary and outlook

1. Introduction Lattice gauge theory gives a nonperturbative definition of the quantum field theory. finite degrees of freedom. ⇒ Monte Carlo simulations ⇒ very powerful tool to study QCD; Hadron spectrum Matrix elements Chiral transition Quark gluon plasma CP-PACS 2002

1. Introduction But the lattice regularization spoils a lot of symmetries… Translational symmetry Lorentz invariance Chiral symmetry or topology Supersymmetry…　

1. Introduction Chiral symmetry Luescher’s admissibility condition, is classically realized by the Ginsparg-Wilson relation. but at quantum level, or in the numerical simulation, D is not well-defined on the topology boundaries. ⇒　crucial obstacle for Nf≠0 overlap fermions. Luescher’s admissibility condition, Improved gauge action which smoothes gauge fields. Additional Wilson fermion action with negative mass. may solve the both theoretical and numerical problems. Ginsparg and Wilson, Phys.Rev.D25,2649(1982) M.Luescher, Nucl.Phys.B538,515(1999) M.Luescher, Private communications

In this work, 1. Introduction we study the “topology conserving actions” in quenched simulation to examine their feasibility; Static quark potential has large scaling violations? Stability of the topological charge ? Numerical cost of the Ginsparg-Wilson fermion ? c.f. W.Bietenholz et al. JHEP 0603:017,2006 .

1. Introduction Monte Carlo simulation of lattice QCD is performed by (Random) small changes of gauge link fields Accept/reject the changes s.t. ～ A classical particle randomly walking in the configuration space.

1. Introduction Example: the hybrid Monte Carlo SU(3) on 20^4 lattice ⇒ a classical particle in a potential S and hit by random force R in 1280000 dimensions. NOTE: each step is small ⇒ tendency of keeping topology.

2. The chiral symmetry and topology Nielsen-Ninomiya theorem: Any local Dirac operator satisfying has unphysical poles (doublers). Example - free fermion – Continuum has no doubler. Lattice has unphysical poles at . Wilson fermion Doublers are decoupled but spoils chiral symmetry. Nielsen and Ninomiya, Nucl.Phys.B185,20(1981)

2. The chiral symmetry and topology Eigenvalue distribution of Dirac operator 2/a 4/a 6/a continuum (massive) m 1/a -1/a

2. The chiral symmetry and topology Wilson fermion Eigenvalue distribution of Dirac operator 2/a 4/a 6/a Naïve lattice fermion 16 lines (massive) m 1/a Doublers are massive. m is not well-defined. The index is not well-defined. -1/a

The Ginsparg-Wilson fermion 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.Lett.B417,141(‘98) Phys.Rev.D25,2649(‘82) M.Luescher,Phys.Lett.B428,342(1998)

2. The chiral symmetry and topology Eigenvalue distribution of Dirac operator 2/a 4/a 6/a The overlap fermion 1/a -1/a D is smooth except for .

2. The chiral symmetry and topology Eigenvalue distribution of Dirac operator 2/a 4/a 6/a The overlap fermion (massive) m 1/a -1/a Doublers are massive. m is well-defined.

2. The chiral symmetry and topology Eigenvalue distribution of Dirac operator 2/a 4/a 6/a The overlap fermion 1/a -1/a Theoretically ill-defined. Large simulation cost.

2. The chiral symmetry and topology The topology (index) changes 2/a 4/a 6/a The complex modes make pairs 1/a Whenever Hw crosses zero, topology changes. The real modes are chiral eigenstates. -1/a

The overlap Dirac operator becomes ill-defined when The topology boundaries. These zero-modes are lattice artifacts(excluded in a→∞limit.) In the polynomial expansion of D, The discontinuity of the determinant requires reflection/refraction (Fodor et al. JHEP0408:003,2004) ~ V2 algorithm.

2. The chiral symmetry and topology SG ε= 1.0 Q=0 　　　　 　 Q=1 If the barrier is high enough, Q may be fixed. SG ε< 1/20 Q=0 　ε=∞ 　 Q=1 The topology conserving gauge action generates configurations satisfying Luescher’s “admissibility” condition: 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… M.Luescher,Nucl.Phys.B568,162 (‘00) M.Creutz, Phys.Rev.D70,091501(‘04) Let’s try larger ε. P.Hernandez et al, Nucl.Phys.B552,363 (1999))

Topological charge is defined as Admissibility in 2D QED Topological charge is defined as Without admissibility condition ⇒　topological charge can jump；　Q → Q±１. If gauge fields are “admissible” ( ε＜ 2 ), ⇒　topological charge is conserved !!

2. The chiral symmetry and topology The negative mass Wilson fermion would also suppress the topology changes. would not affect the low-energy physics.

3. Lattice simulations 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 Topology conserving gauge 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) 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. The simulations were done on the Alpha work station at YITP and SX-5 at RCNP.

3. Lattice simulations Negative mass Wilson fermion With s=0.6. Topology conserving gauge action (1/ε=1,2/3,0) Algorithm: HMC + pseudofermion Lattice size : 144,164 . 1 trajectory = 15 - 60 molecular dynamics steps with stepsize Δτ= 0.007-0.01. size 1/ε β Δτ Nmds acceptance Plaquette 144 1.0 0.75 0.01 15 72% 0.52260(2) 2/3 1.8 87% 0.52915(3) 0.0 5.0 88% 0.55377(6) 164 0.8 0.007 60 79% 0.53091(1) 1.75 0.008 50 89% 0.52227(4) 5.2 93% 0.57577(3) The simulations were done on the Alpha work station at YITP and SX-5 at RCNP.

3. Lattice simulations Implementation of the overlap 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. Eigenvalues are calculated with ARPACK, too. ARPACK, available from http://www.caam.rice.edu/software/

3. Lattice simulations Initial configuration 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)

3. Lattice simulations New cooling method to measure Q 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) 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 .

4. Results The static quark potential With det Hw2 quenched 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 ref S.Necco,R.Sommer,Nucl.Phys.B622,328(‘02) Sommer scales are determined by

4. Results 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 ref S.Necco,R.Sommer,Nucl.Phys.B622,328(‘02) Sommer scales are determined by

Continuum limit (Necco,Sommer ‘02) 4. Results quenched The static quark potential Here we assume r0 ~ 0.5 fm. 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)

Continuum limit (Necco,Sommer ‘02) 4. Results With det Hw2 (Preliminary) The static quark potential size 1/ε β samples r0/a rc/a a rc/r0 164 1.0 0.8 153 5.12(61) 2.473(51) ~0.10fm 0.483(56) 2/3 1.75 145 4.63(29) 2.307(60) ~0.11fm 0.498(34) 5.2 225 7.09(17) 3.462(55) ~0.07fm 0.489(13) 144 0.75 162 4.24(15) 2.240(37) ~0.12fm 0.528(24) 1.8 261 4.94(19) 2.361(26) 0.478(19) 5.0 4.904(90) 2.691(42) 0.549(13) Continuum limit (Necco,Sommer ‘02) 0.5133(24)

4. Results Renormalization of the coupling The renormalized coupling in Manton-scheme is defined where is the tadpole improved bare coupling: where P is the plaquette expectation value. quenched R.K.Ellis,G.Martinelli, Nucl.Phys.B235,93(‘84)Erratum-ibid.B249,750(‘85)

4. Results The stability of the topological charge The stability of Q for 4D QCD is proved only when ε＜ εmax ～1/20 ,which is not practical… Topology preservation should be perfect.

4. Results The stability of the topological charge 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. M.Luescher, hep-lat/0409106 Appendix E.

quenched 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 quenched

4. Results Topology conservation seems perfect ! With det Hw2 size 1/ε β r0/a Trj τplaq #Q Q stability 164 1.0 0.8 5.12(61) 480 0.65(1) >693 2/3 1.75 4.63(29) 454 1.8(5) >483 0.0 5.2 7.09(17) 730 1.5(3) >146 144 0.75 4.24(15) 3500 5.1(8) >741 1.8 4.94(19) 5370 11(2) >251 5.0 4.904(90) 3120 21(6) >474 Topology conservation seems perfect !

4. Results Numerical cost of overlap Dirac operator We expect Low-modes of Hw are suppressed. ⇒ the Chebyshev approximation is improved. : The condition number of Hw : order of polynomial : constants independent of V, β, ε… Locality is improved.

4. Results The eigenvalues of |Hw| The admissibility condition plaquette 1/ε=2/3 1/ε=1 plaquette with det Hw2 r0~6.5-7 The eigenvalues of |Hw| The admissibility condition ⇒　pushes up the average of low-eigenvalues of |Hw|. (the gain ~ 2-3 factors.) det Hw2 (Negative mass Wilson fermion) ⇒　the very small eigenvalues (<<0.1) are suppressed.

4. Results The locality For should exponentially decay. 1/a~0.08fm (with 4 samples), no remarkable improvement of locality is seen… ⇒　lower beta? quenched ＋ : beta = 1.42, 1/e=1.0 × : beta = 2.7, 1/e=2/3 ＊ : beta = 6.13, 1/e=0.0

How to sum up the different topological sectors ⇒　We need 　　 .

How to sum up the different topological sectors Formally, With an assumption, The ratio can be given by the topological susceptibility, if it has small Q and V’ dependences. Parallel tempering + Fodor method may also be useful. V’ Z.Fodor et al. hep-lat/0510117

Topology dependence If , any observable at a fixed topology in general theory (with θvacuum) can be written as Brower et al, Phys.Lett.B560(2003)64 In QCD, ⇒ Unless ,（like NEDM） Q-dependence is negligible. Shintani et al,Phys.Rev.D72:014504,2005

5. Summary and Outlook The overlap Dirac operator, realizes the exact chiral symmetry at classical level. However, at quantum level, the topology boundary, should be excluded for sound construction of quantum field theory. numerical cost down. Topology conserving actions; Keeping the “admissibility” condition: Negative mass Wilson fermions: can be helpful to suppress Hw~0 when 1/a~2-3GeV.

5. Summary and Outlook We have studied ‘Topology conserving actions’ in the pure gauge SU(3) theory. The Wilson loops show no large O(a) effects. Admissibility condition does not induce large scaling violation. negative mass Wilson fermions are decoupled. Q can be fixed. (100-1000 uncorrelated samples ) Small Hw is suppressed.

5. Summary and Outlook Better choice ? Including twisted mass ghost, would cancel the higher mode contributions. -> smaller scaling violations. would require cheaper numerical cost. Converges to 1 in the continuum limit with mt fixed.

5. Summary and Outlook For future works, we would like to try Summation of different topology Nf=2 overlap fermion with fixed topology Full QCD in the epsilon-regime. Hadron spectrum, decay constants, chiral condensates… Finite temperature θ vacuum Supersymmetry…

(Pion mass)2 vs quark mass 6. Nf=2 lattice QCD at KEK (Pion mass)2 vs quark mass [preliminary] Cost of GW fermion ~ Naively 100 times larger than Wilson fermion. Or much more for non-smooth determinant. KEK BlueGene (started on March 1st) is 50 times faster ! Our topology conserving determinant (with twisted mass ghost) is adopted. Now test run with Iwasaki gauge action + 2-flavor overlap fermions and topology conserving determinant on 16332 is underway. First result with exact chiral symmetric Dirac operator is coming soon.

6. Nf=2 lattice QCD at KEK Old JLQCD collaboration L~2fm, mu=md>50MeV. Nf=2+1 Wilson fermion + O(a) improvement term New JLQCD test RUN (from March 2006) L~2fm, mu=md > 2MeV . Nf=2 Ginsparg-Wilson fermion Confugurations are Q=0 sector only. Future plan L~3fm, mu=md>10-20MeV Nf=2+1 Ginsparg-Wilson fermion

6. Nf=2 lattice QCD at KEK Cost of GW fermion ~ Naively 100 times larger than Wilson fermion. Or much more for non-smooth determinant. KEK BlueGene (started on March 1st) is 50 times faster ! Our topology conserving determinant (with twisted mass ghost) is adopted. Now test run with Iwasaki gauge action + 2-flavor overlap fermions and topology conserving determinant on 16332 is underway. First result with exact chiral symmetric Dirac operator is coming soon.

4. Results Q dependence of the quark potential seems week Topology dependence 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)

4. Results The condition number quenched 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) 1.42 6.240(89) 63 0.0282(21) 0.031(10) 2.7 6.559(76) 110 0.0251(19) 0.019(18) 6.13 6.642(-) 35 0.0126(15) 0.084(14) 164 961 0.0367(21) 0.007(5) 752 0.0320(19) 0.020(8) 20 0.0232(17) 0.030(10)

4. Results The condition number With det Hw2 (Preliminary) 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) 2/3 1.75 6.26(36) >46 0.1284(13) 0.08 0.0 5.2 6.16(19) >32 0.2325(17) 0.05 quenched 6.13 6.642 20 0.139(10) 0.03