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

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

1. Introduction
2. The overlap fermion and topology
3. Lattice simulations
4. Results
5. 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 Neubergers 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 ? Lets try !

c.f. W.Bietenholz et al. hep-lat/0511016.
9
2. The overlap fermion and topology

• Eigenvalue distribution of Dirac operators

1/a
-1/a
10
2. The overlap fermion and topology

• Eigenvalue distribution of Dirac operators

1/a
-1/a

• Doublers are massive.
• m is not well-defined.

11
2. The overlap fermion and topology

• Eigenvalue distribution of Dirac operators

1/a
-1/a
12
2. The overlap fermion and topology

• Eigenvalue distribution of Dirac operators

1/a
-1/a

• Doublers are massive.
• m is well-defined.

13
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
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 ? are constants.
• Numerical cost
• In the polynomial approximation for D
• The discontinuity of the determinant requires
• reflection/refraction (Fodor et al.
  JHEP0408003,2004)

16
2. The overlap fermion and topology

• The topology conserving gauge action
• generates configurations satisfying the
  admissibility
• bound
• NOTE
• The effect of e is O(a4) and the positivity is
  restored as
• e/a4 ? 8.
• Hw gt 0 if e lt 1/20.49, but it s too small

M.Luescher,Nucl.Phys.B568,162 (00)
M.Creutz, Phys.Rev.D70,091501(04)
Lets try larger e.
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
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
size 1/e ß ?t Nmds acceptance Plaquette
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)

• Topology conserving gauge action (quenched)
• with 1/e 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 ?t 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 s0.6.
• Topology conserving gauge action (1/e1,2/3,0)
• Algorithm HMC pseudofermion
• Lattice size 144,164 .
• 1 trajectory 10 - 15 molecular dynamics steps
• with stepsize ?t 0.01.

size 1/e ß ?t 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)
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/so
ftware/
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 ß106, (after
  40-50
• steps) then
• gives a number close to the index of the overlap
  operator.
• NOTE 1/ecool 2/3 is useful for 1/e 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/e 1.0 and 2/3.
• 60-70 for 1/e0.0 (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 Q0 )
• We measure the Wilson loops, in
• 6 different spatial direction,
• using smearing. G.S.Bali,K.Schilling,Phys.R
  ev.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 Q0 )
• We measure the Wilson loops, in
• 6 different spatial direction,
• using smearing. G.S.Bali,K.Schilling,Phys.R
  ev.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.

size 1/e ß samples r0/a rc/a a rc/r0
124 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) Continuum limit (Necco,Sommer 02) Continuum limit (Necco,Sommer 02) Continuum limit (Necco,Sommer 02) Continuum limit (Necco,Sommer 02) Continuum limit (Necco,Sommer 02) 0.5133(24)
29
4. Results
With det Hw2 (Preliminary)

• The static quark potential

size 1/e ß samples r0/a rc/a a rc/r0
164 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) Continuum limit (Necco,Sommer 02) Continuum limit (Necco,Sommer 02) Continuum limit (Necco,Sommer 02) Continuum limit (Necco,Sommer 02) Continuum limit (Necco,Sommer 02) 0.5133(24)
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)Erra
tum-ibid.B249,750(85)
31
4. Results

• The stability of the topological charge
• The stability of Q for 4D QCD is proved only when
  
• elt emax 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
size 1/e ß r0/a Trj tplaq Q Q stability
124 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 5
0.0 5.8 3.668(12) 18205 30.2(6.6) 728 1
1.0 1.2 4.464(65) 18000 1.59(15) 265 43
2/3 2.4 4.390(99) 18000 2.62(23) 400 17
0.0 5.9 4.483(17) 27116 13.2(1.5) 761 3
1.0 1.3 5.240(96) 18000 1.091(70) 69 239
2/3 2.55 5.290(69) 18000 2.86(33) 123 51
0.0 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 46
2/3 2.55 5.290(69) 12000 6.4(5) 107 18
0.0 6.0 5.368(22) 3500 11.7(3.9) 166 1.8
1.0 1.42 6.240(89) 5000 2.6(4) 2 961
2/3 2.7 6.559(76) 14000 3.1(3) 6 752
0.0 6.13 6.642(-) 5500 12.4(3.3) 22 20
204 1.0 1.3 5.240(96) 1240 2.6(5) 14 34
2/3 2.55 5.290(69) 1240 3.4(7) 15 24
0.0 6.0 5.368(22) 1600 14.4(7.8) 37 3
1.0 1.42 6.240(89) 7000 3.8(8) 29 63
2/3 2.7 6.559(76) 7800 3.5(6) 20 110
0.0 6.13 6.642(-) 1298 9.3(2.8) 4 35
quenched
34
4. Results
With det Hw2 (Preliminary)
size 1/e ß r0/a Trj tplaq Q Q stability
164 1.0 0.8 5.7(1.0) 520 12(5) 0 gt43
2/3 1.75 6.26(36) 460 10(4) 0 gt46
0.0 5.2 6.16(19) 1614 51(31) 0 gt32
144 1.0 0.75 4.97(58) 560 5(2) 0 gt112
2/3 1.8 5.68(90) 1360 14(5) 0 gt97
0.0 5.0 6.1(1.2) 480 11(5) 0 gt44
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, ß, e
• Locality is improved.

36
4. Results

• The condition number
• The gain is about a factor 2-3.

quenched
size 1/e ß r0/a Q stability 1/? P(lt0.1)
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)
37
4. Results
With det Hw2 (Preliminary)

• The condition number

size 1/e ß r0/a Q stability hwmin P(lt0.1)
164 1.0 0.8 5.7(1.0) gt43 0.1823(33) 0
2/3 1.75 6.26(36) gt46 0.1284(13) 0.08
0.0 5.2 6.16(19) gt32 0.2325(17) 0.05
quenched 0 6.13 6.642 20 0.139(10) 0.03
38
4. Results

• The locality
• For
• should exponentially decay.
• 1/a0.08fm
• (with 4 samples),
• no remarkable
• improvement of
• locality is seen
• ? lower beta?

quenched
beta 1.42, 1/e1.0 beta 2.7,
1/e2/3 beta 6.13, 1/e0.0
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.

size 1/e ß Initial Q Q stability plaquette r0/a rc/r0
164 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)