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

2. Contents Introduction Lattice simulations The static quark potential
Stability of the topological charge
The overlap Dirac operator
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. Nielsen-Ninomiya theorem
Any local Dirac operator satisfying chiral symmetry;
has unphysical poles (doublers).
Example – free naïve fermion –
Continuum has no doubler.
Lattice
has unphysical poles at
Wilson Dirac operator
Doublers are decoupled (heavy) but no chiral symmetry.
Nucl.Phys.B185,20 (‘81),Nucl.Phys.B193,173 (‘81)

5. locality? or smoothness?
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
locality? or smoothness?
Phys.Lett.B417,141(‘98)
Phys.Rev.D25,2649(‘82)

6. The locality of the overlap operator If the gauge fields are smooth;
with a (small) fixed number ε , it gives a lower bound to
⇒ (exponential) locality and smoothness of the Dirac operator;
⇒ Stability of the index (topological charge);
P.Hernandez,K.Jansen,M.Luescher,Nucl.Phys.B552,363(‘99)

7. Topology conserving gauge action
The bound is
automatically satisfied with the action
Hybrid Monte Carlo (HMC) algorithm
Heat bath algorithm = local and large updates
HMC algorithm = global and small updates
⇒　Q may be fixed along the simulations
M.Luescher,Phys.Lett.B428,342(‘98),Nucl.Phys.B549,295(‘99)

8. - Motivations -
We studied the Nf=2 massive Schwinger model
with a theta term using the topology conserving action and the domain-wall fermion action;
HF,T.Onogi,Phys.Rev.D68,074503(‘03),Phys.Rev.D70,054508(’04)
Good chiral behavior (consistent with the
bosonization approach).
Perfect stability of Q with ε= 1.0.
Efficient sampling of the configurations with fixed Q ⇒ ε- regime with fixed Q
Improvement of the locality of the overlap operator.
Lower numerical costs of the overlap operator
⇒　dynamical overlap fermions
Anyway, it’s interesting…

9. - Our goals - Scaling studies (to show validity in 4D quenched QCD.)
determination of the lattice spacing
scaling violations of the quark potential
How stable Q ?
Locality and numerical cost of GW fermion improved ?
c.f. S.Shcheredin, W.Bietenholz, K.Jansen, K.I.Nagai, S.Necco and L.Scorzato, hep-lat/ , hep-lat/

10. 2. Lattice simulations Action: Luescher 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)
2. Lattice simulations
Action: Luescher 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.

11. The overlap Dirac 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.
We set s=0.6
ARPACK, available from

12. 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)

13. 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 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 .

14. Cooling -Example: 2D QED-
- instanton anti-instanton pair on 162 lattice -
Eigen function （chirality －）
Eigen function （chirality +）
Eigen values of γ５D
Let’s cool it !

15. Cooling -Example: 2D QED-
- 2 instanton and 1 anti-instanton -
→1 instanton
Eigenvalues
Let’s cool it !

16. Cooling -4D QCD results-
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)

17. 3. 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 S.Necco,R.Sommer,Nucl.Phys.B622,328(‘02)
Sommer scales are determined by

18. Continuum limit (Necco,Sommer ‘02)
Assuming r0 ~ 0.5 fm, we obtain
(preliminary)
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)

19. The quark potential itself is good.

20. Perturbative renormalization of the coupling
At 1-loop level, renormalized couplings have similar values but the convergence of the perturbative series is much worse for 1/ε= 1, 2/3.
Tadpole improvement at 2-loop level may be required or non-perturbative approach should be done.
R.K.Ellis,G.Martinelli,
Nucl.Phys.B235,93(‘84)
Erratum-ibid.B249,750(‘85)

21. 4. Stability of the topological chargeSG
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
Let us search the parameters;
β,　ε,　V which can fix Q.
The stability of Q for 4D QCD is proved only when
ε＜ εmax ～1/30 ,which is not practical…

22. We define “stability” by the ratio of topology change
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,
since there may be topology changes during every 20
trajectories.
M.Luescher, hep-lat/ Appendix E.

23. Stability is better for smaller ε, smaller L , smaller a.
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
Stability is better for smaller ε, smaller L , smaller a.
A significantly good stability can be seen
for
⇒　maybe valid in ε- regime.

24. 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)

25. 5. The overlap Dirac operator
It is known that the lowest eigen value of is related to
Locality is guaranteed if never touches zero.
The order of Chebyshev polynomial can be reduced;
: the condition number
: a random vector

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

27. We use 10 different Npol and 4 different confs for each parameter set.
The condition number and Chebyshev approximation
We use 10 different Npol and 4 different confs for each parameter set.
size
1/ε β
r0/a
Q stability
204
1.0
1.3
5.240(96) 34
0.0148(14)
(29)
2/3
2.55
5.290(69) 24
0.0101(08)
(27)
0.0
6.0
5.368(22) 3
0.0059(34)
(46)
1.42
6.240(89) 63
0.0282(21)
(32)
2.7
6.559(76)
110
0.0251(19)
(37)
6.13
6.642(-) 35
0.0126(15)
(50)
164
961
0.0367(21)
(26)
752
0.0320(19)
(29) 20
0.0232(17)
(38)
and are nor independent.
For 1/ε=1.0, is 1.6~2.5 times larger
( is 1.2~1.4 times larger) than that with the
plaquette action.

28. 6. Conclusion and outlook
Topology conserving gauge action may be practical.
New cooling method does work.
The lattice spacing can be determined in a conventional manner, ant the quark potential show no large deviation from the continuum limit.
Q can be stable for .
No improvement of the locality (for high beta).
The numerical cost of Chebyshev approximation would be times better than that with plaquette action.

29. Outlooks
Accept/reject for topology changes to fix Q completely (per every 100trj would be enough.).
Quenched ε-regime with fixed Q.
Nf=2 full QCD in the ε-regime with the overlap fermion.