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Topology conserving actions and the overlap Dirac operator (hep-lat/0510116)

Topology conserving actions and the overlap Dirac operator (hep-lat/0510116). Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto (KEK,Sokendai), T.Hirohashi (Kyoto Univ.), H.Matsufuru(KEK), K.Ogawa(Sokendai) and T.Onogi(YITP). Contents. Introduction

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Topology conserving actions and the overlap Dirac operator (hep-lat/0510116)

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  1. Topology conserving actions and the overlap Dirac operator (hep-lat/0510116) Hidenori Fukaya Yukawa Institute, Kyoto Univ. Collaboration with S.Hashimoto (KEK,Sokendai), T.Hirohashi (Kyoto Univ.), H.Matsufuru(KEK), K.Ogawa(Sokendai) and T.Onogi(YITP)

  2. Contents • Introduction • The overlap fermion and topology • Lattice simulations • Results • Conclusion and outlook

  3. 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…

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

  5. 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.Lett.B417,141(‘98) Phys.Rev.D25,2649(‘82)

  6. 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.

  7. Topology conserving actions can be achieved by • The “admissibility” condition • The determinant (The negative mass Wilson fermion) Details are in the next section…

  8. 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.

  9. continuum (massive) 0 2/a 4/a 6/a m 2. The overlap fermion and topology • Eigenvalue distribution of Dirac operators 1/a -1/a

  10. Wilson fermion naïve fermion (massive) 16 lines 0 2/a 4/a 6/a m 2. The overlap fermion and topology • Eigenvalue distribution of Dirac operators 1/a -1/a • Doublers are massive. • m is not well-defined.

  11. The overlap fermion 0 2/a 4/a 6/a • D is smooth except for . 2. The overlap fermion and topology • Eigenvalue distribution of Dirac operators 1/a -1/a

  12. The overlap fermion (massive) 0 2/a 4/a 6/a m 2. The overlap fermion and topology • Eigenvalue distribution of Dirac operators 1/a -1/a • Doublers are massive. • m is well-defined.

  13. The overlap fermion 0 2/a 4/a 6/a 2. The overlap fermion and topology • Eigenvalue distribution of Dirac operators 1/a -1/a • Topology boundary. • Locality may be lost. • Large simulation cost.

  14. The complex modes make pairs 0 2/a 4/a The real modes are chiral eigenstates. 6/a 2. The overlap fermion and topology • The topology (index) changes 1/a -1/a

  15. 2. The overlap fermion and topology • 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)

  16. 2. The overlap fermion and topology • The topology conserving gauge action generates configurations satisfying the “admissibility” bound: NOTE: • The effect of εis 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.

  17. 2. The overlap fermion and topology • 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…

  18. ⇒We need . 2. The overlap fermion and topology • How to sum up the different topological sectors

  19. 2. The overlap fermion and topology • 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. V’ Z.Fodor et al. hep-lat/0510117

  20. 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

  21. 3. Lattice simulations • 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.

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

  23. 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/

  24. 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)

  25. 3. Lattice simulations • 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 . 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)

  26. 4. Results With det Hw2 (Preliminary) 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

  27. 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

  28. 4. Results quenched • The static quark potential Here we assume r0 ~ 0.5 fm.

  29. 4. Results With det Hw2 (Preliminary) • The static quark potential

  30. 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)

  31. 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 4. Results • 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 perfect But large scaling violations??

  32. 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.

  33. quenched

  34. 4. Results With det Hw2 (Preliminary) Topology conservation seems perfect !

  35. 4. Results • The overlap Dirac operator We 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.

  36. 4. Results • The condition number The gain is about a factor 2-3. quenched

  37. 4. Results With det Hw2 (Preliminary) • The condition number

  38. 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

  39. 5. Conclusion and Outlook We find • New cooling method does work. • In quenched study, the lattice spacing can be determined in a conventional 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.

  40. 5. Conclusion and Outlook For future works, we would like to try • Including twisted mass ghost, • Summation of different topology • Dynamical overlap fermion at fixed topology

  41. 4. Results • Topology dependence Q dependence of the quark potential seems week as we expected.

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