Lattice%20QCD%20in%20a%20fixed%20topological%20sector%20[hep-lat/0603008]

1
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)

2
Contents

1. Introduction
2. The chiral symmetry and topology
3. Lattice simulations
4. Results
5. Summary and outlook

3
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



4
1. Introduction

• But the lattice regularization spoils a lot of
  symmetries
• Translational symmetry
• Lorentz invariance
• Chiral symmetry or topology
• Supersymmetry

5
1. Introduction

• Chiral symmetry
• 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.
• Lueschers 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
6
1. Introduction

• In this work,
• 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 0603017,2006 .
7
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.

8
1. Introduction

• Example the hybrid Monte Carlo
• SU(3) on 204 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.

9
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)
10
2. The chiral symmetry and topology

• Eigenvalue distribution of Dirac operator

1/a
-1/a
11
2. The chiral symmetry and topology

• Eigenvalue distribution of Dirac operator

1/a

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

-1/a
12

• The Ginsparg-Wilson fermion
• 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)
M.Luescher,Phys.Lett.B428,342(1998)
13
2. The chiral symmetry and topology

• Eigenvalue distribution of Dirac operator

1/a
-1/a
14
2. The chiral symmetry and topology

• Eigenvalue distribution of Dirac operator

1/a
-1/a

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

15
2. The chiral symmetry and topology

• Eigenvalue distribution of Dirac operator

1/a
-1/a

• Theoretically ill-defined.
• Large simulation cost.

16
2. The chiral symmetry and topology

• The topology (index) changes

1/a
Whenever Hw crosses zero, topology changes.
-1/a
17

• The overlap Dirac operator
• becomes ill-defined when
• The topology boundaries.
• These zero-modes are lattice artifacts(excluded
  in a?8limit.)
• In the polynomial expansion of D,
• The discontinuity of the determinant requires
• reflection/refraction (Fodor et al.
  JHEP0408003,2004)
• V2 algorithm.

18
2. The chiral symmetry and topology

• The topology conserving gauge action
• generates configurations satisfying Lueschers
  admissibility condition
• 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.
P.Hernandez et al, Nucl.Phys.B552,363 (1999))
19

• Admissibility in 2D QED
• Topological charge is defined as

20
2. The chiral symmetry and topology

• The negative mass Wilson fermion
• would also suppress the topology changes.
• would not affect the low-energy physics.

21
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

22
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.
23
3. Lattice simulations

• Negative mass Wilson fermion
• With s0.6.
• Topology conserving gauge action (1/e1,2/3,0)
• Algorithm HMC pseudofermion
• Lattice size 144,164 .
• 1 trajectory 15 - 60 molecular dynamics steps
• with stepsize ?t 0.007-0.01.

size 1/e ß ?t Nmds acceptance Plaquette
144 1.0 0.75 0.01 15 72 0.52260(2)
2/3 1.8 0.01 15 87 0.52915(3)
0.0 5.0 0.01 15 88 0.55377(6)
164 1.0 0.8 0.007 60 79 0.53091(1)
2/3 1.75 0.008 50 89 0.52227(4)
0.0 5.2 0.008 50 93 0.57577(3)
The simulations were done on the Alpha work
station at YITP and SX-5 at RCNP.
24
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/
25
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)
26
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)

27
4. Results
With det Hw2
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

28
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

29
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)
30
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 0.8 153 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)
0 5.2 225 7.09(17) 3.462(55) 0.07fm 0.489(13)
144 1.0 0.75 162 4.24(15) 2.240(37) 0.12fm 0.528(24)
2/3 1.8 261 4.94(19) 2.361(26) 0.10fm 0.478(19)
0 5.0 162 4.904(90) 2.691(42) 0.10fm 0.549(13)
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)
31
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)
32
4. Results

• The stability of the topological charge
• The stability of Q for 4D QCD is proved only when
  
• elt emax 1/20 ,which is not practical
• Topology preservation should be perfect.

33
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.
34
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
35
4. Results
With det Hw2
size 1/e ß r0/a Trj tplaq Q Q stability
164 1.0 0.8 5.12(61) 480 0.65(1) 0 gt693
2/3 1.75 4.63(29) 454 1.8(5) 0 gt483
0.0 5.2 7.09(17) 730 1.5(3) 0 gt146
144 1.0 0.75 4.24(15) 3500 5.1(8) 0 gt741
2/3 1.8 4.94(19) 5370 11(2) 0 gt251
0.0 5.0 4.904(90) 3120 21(6) 0 gt474
Topology conservation seems perfect !
36
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, ß, e
• Locality is improved.

37
4. Results

• 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 (ltlt0.1) are
  suppressed.

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

• How to sum up the different topological sectors

40

• 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
41

• Topology dependence
• If , any observable at a fixed topology
  in general theory (with ?vacuum) 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.D72014504,2005
42
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 Hw0 when 1/a2-3GeV.

43
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.

44
5. Summary and Outlook

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

45
5. Summary and Outlook

• For future works, we would like to try
• Summation of different topology
• Nf2 overlap fermion with fixed topology
• Full QCD in the epsilon-regime.
• Hadron spectrum, decay constants, chiral
  condensates
• Finite temperature
• ? vacuum
• Supersymmetry

46
6. Nf2 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.

47
6. Nf2 lattice QCD at KEK

• Old JLQCD collaboration
• L2fm, mumdgt50MeV.
• Nf21 Wilson fermion O(a) improvement term
• New JLQCD test RUN (from March 2006)
• L2fm, mumd gt 2MeV .
• Nf2 Ginsparg-Wilson fermion
• Confugurations are Q0 sector only.
• Future plan
• L3fm, mumdgt10-20MeV
• Nf21 Ginsparg-Wilson fermion

48
6. Nf2 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.

49
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)
50
4. Results

• The condition number

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)
51
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