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

2. Contents Introduction The chiral symmetry and topology
Lattice simulations
Results
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 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

6. 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 .

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 20^4 lattice ⇒
a classical particle in a potential S
and hit by random force R in 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
2/a
4/a
6/a
continuum
(massive) m
1/a
-1/a

11. 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

12. 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)

13. 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

14. 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.

15. 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.

16. 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

17. 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.

18. 2. The chiral symmetry and topologySG
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))

19. 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 !!

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 Topology conserving gauge action (quenched)
size
1/ε β Δτ
Nmds
acceptance
Plaquette
124
1.0
0.01 40
89%
(9)
1.2
90%
(6)
1.3
(6)
2/3
2.25
93%
(1)
2.4
(1)
2.55
(1)
0.0
5.8
0.02 20
69%
(5)
5.9
(3)
6.0
68%
(2)
164
82%
(1)
1.42
(1)
88%
(2)
2.7
87%
(1)
(5)
6.13
(4)
204
72%
(9)
74%
(1)
(2)
(1)
0.015
53%
(4)
83%
(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 = molecular dynamics steps
with stepsize Δτ=
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 s=0.6.
Topology conserving gauge action (1/ε=1,2/3,0)
Algorithm: HMC + pseudofermion
Lattice size : 144,164 .
1 trajectory = molecular dynamics steps
with stepsize Δτ=
size
1/ε β Δτ
Nmds
acceptance
Plaquette
144
1.0
0.75
0.01 15
72%
(2)
2/3
1.8
87%
(3)
0.0
5.0
88%
(6)
164
0.8
0.007 60
79%
(1)
1.75
0.008 50
89%
(4)
5.2
93%
(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

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/ , M.Hamanaka,H.Kajiura,Phys.Lett.B551,360(‘03)

26. 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 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 .

27. 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

28. 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

29. 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)

30. 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)

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)Erratum-ibid.B249,750(‘85)

32. 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.

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/ Appendix E.

34. 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

35. 4. Results Topology conservation seems perfect ! With det Hw2 size 1/εβ
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 !

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, β, ε…
Locality is improved.

37. 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.

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

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/

41. 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

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 Hw~0 when 1/a~2-3GeV.

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. ( uncorrelated samples )
Small Hw is suppressed.

44. 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.

45. 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…

46. (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 is underway.
First result with exact chiral symmetric Dirac operator is coming soon.

47. 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

48. 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 is underway.
First result with exact chiral symmetric Dirac operator is coming soon.

49. 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
(1)
6.240(89)
0.5126(98) -3
514
(1)
6.11(13)
0.513(12)

50. 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)

51. 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