topology conserving actions and the overlap dirac operator (hep-lat/0510116) hidenori fukaya yukawa...

Topology conserving actions and the overlap Topology conserving actions and the overlap Dirac operatorDirac operator

(hep-lat/0510116) (hep-lat/0510116) Hidenori FukayaYukawa Institute, Kyoto Univ.

Collaboration with S.Hashimoto (KEK,Sokendai), T.Hirohashi (Kyoto Univ.),H.Matsufuru(KEK), K.Ogawa(Sokendai) and T.Onogi(YITP)

Contents

1. Introduction2. The overlap fermion and topology3. Lattice simulations4. Results5. Conclusion and outlook

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… 　

1. Introduction

Nielsen-Ninomiya theorem

Any local Dirac operator satisfying chiral symmetry has unphysical poles (doublers).

Example - free fermion – Continuum has no double ｒ . Lattice

has unphysical poles at . Wilson fermion

Doublers are decoupled but no chiral symmetry.

Nucl.Phys.B185,20 (‘81),Nucl.Phys.B193,173 (‘81)

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.Rev.D25,2649(‘82)

Phys.Lett.B417,141(‘98)

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.

Topology conserving actions

can be achieved by

The “admissibility” condition

The determinant (The negative mass Wilson fermion)

Details are in the next section…

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.

02/a 4/a 6/a

Eigenvalue distribution of Dirac operators

2. The overlap fermion and topology

continuum

(massive)

m

1/a

-1/a

Wilson fermion



1/a

-1/a

naïve fermion

16 lines

02/a 4/a 6/a

(massive)

m

• Doublers are massive.

• m is not well-defined.



1/a

-1/a

02/a 4/a 6/a

The overlap fermion

• D is smooth except for .



1/a

-1/a

02/a 4/a 6/a

The overlap fermion(massive)

m

• Doublers are massive.

• m is well-defined.



1/a

-1/a

02/a 4/a 6/a

The overlap fermion

• Topology boundary.

• Locality may be lost.

• Large simulation cost.

The topology (index) changes


1/a

-1/a

02/a 4/a 6/a

The complex modes make pairs

The real modes are chiral eigenstates.

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)


The topology conserving gauge action

generates configurations satisfying the “admissibility” bound: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.Creutz, Phys.Rev.D70,091501(‘04)

M.Luescher,Nucl.Phys.B568,162 (‘00)

Let’s try larger ε.

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…


How to sum up the different topological sectors


⇒ 　 We need 　　　　 ..

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

In this talk,

Topology conserving gauge action (quenched)

Negative mass Wilson fermion

Future works …

Summation of different topology Dynamical overlap fermion at fixed topology

3. Lattice simulations

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

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.

3. Lattice simulationssize 1/ε β Δτ Nmds acceptanc

ePlaquette

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)

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.

size 1/ε β Δτ 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)

Implementation of the overlap operator

We use the implicit restarted Arnoldi method (ARPACK) to calculate the eigenvalues of .

To compute , we use the Chebyshevpolynomial approximation after subtracting 10 lowest eigenmodes exactly.

Eigenvalues are calculated with ARPACK, too.


ARPACK, available from http://www.caam.rice.edu/software/

Initial configurationFor 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)

New cooling method to measure QWe “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 ofoverlap 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)

The static quark potentialIn the following, we assume Q does not affect the Wilson loops. ( initial Q=0 )

1. We measure the Wilson loops, in6 different spatial direction,

using smearing. G.S.Bali,K.Schilling,Phys.Rev.D47,661(‘93)

2. The potential is extracted as .

3. From results, we calculate the force following ref S.Necco,R.Sommer,Nucl.Phys.B622,328(‘02)

4. Sommer scales are determined by

4. Results quenchedWith det Hw2

(Preliminary)

The static quark potentialIn the following, we assume Q does not affect the Wilson loops. ( initial Q=0 )

1. We measure the Wilson loops, in6 different spatial direction,

using smearing. G.S.Bali,K.Schilling,Phys.Rev.D47,661(‘93)

2. The potential is extracted as .

3. From results, we calculate the force following ref S.Necco,R.Sommer,Nucl.Phys.B622,328(‘02)

4. Sommer scales are determined by

4. Results

The static quark potential

Here we assume r0 ~ 0.5 fm.

4. Results

size 1/ε β samples r0/a rc/a a rc/r0124 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) 0.5133(24)

quenched

The static quark potential

4. Results

size 1/ε β samples r0/a rc/a a rc/r0164 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) 0.5133(24)

With det Hw2 (Preliminary)

Renormalization of the couplingThe renormalized coupling in Manton-scheme is defined

where is the tadpole improved bare coupling:

where P is the plaquette expectation value.

4. Results

R.K.Ellis,G.Martinelli, Nucl.Phys.B235,93(‘84)Erratum-ibid.B249,750(‘85)

quenched

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 perfectBut large scaling violations??

4. Results

SG

ε< 1/30

Q=0 　 ε=∞ 　 Q=1

SG

ε= 1.0

Q=0 　　　　　 Q=1

If the barrier is high enough, Q may be fixed.

The stability of the topological chargeWe 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.

4. Results

M.Luescher, hep-lat/0409106 Appendix E.

size 1/ε β r0/a Trj τplaq #Q Q stability124 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 50.0 5.8 [3.668(12)] 18205 30.2(6.6) 728 11.0 1.2 4.464(65) 18000 1.59(15) 265 432/3 2.4 4.390(99) 18000 2.62(23) 400 170.0 5.9 [4.483(17)] 27116 13.2(1.5) 761 31.0 1.3 5.240(96) 18000 1.091(70) 69 2392/3 2.55 5.290(69) 18000 2.86(33) 123 510.0 6.0 [5.368(22)] 27188 15.7(3.0) 304 6

164 1.0 1.3 5.240(96) 11600 3.2(6) 78 462/3 2.55 5.290(69) 12000 6.4(5) 107 180.0 6.0 [5.368(22)] 3500 11.7(3.9) 166 1.81.0 1.42 6.240(89) 5000 2.6(4) 2 9612/3 2.7 6.559(76) 14000 3.1(3) 6 7520.0 6.13 [6.642(-)] 5500 12.4(3.3) 22 20

204 1.0 1.3 5.240(96) 1240 2.6(5) 14 342/3 2.55 5.290(69) 1240 3.4(7) 15 240.0 6.0 [5.368(22)] 1600 14.4(7.8) 37 31.0 1.42 6.240(89) 7000 3.8(8) 29 632/3 2.7 6.559(76) 7800 3.5(6) 20 1100.0 6.13 [6.642(-)] 1298 9.3(2.8) 4 35

quenched

size 1/ε β r0/a Trj τplaq #Q Q stability164 1.0 0.8 5.7(1.0) 520 12(5) 0 >43

2/3 1.75 6.26(36) 460 10(4) 0 >460.0 5.2 6.16(19) 1614 51(31) 0 >32

144 1.0 0.75 4.97(58) 560 5(2) 0 >1122/3 1.8 5.68(90) 1360 14(5) 0 >970.0 5.0 6.1(1.2) 480 11(5) 0 >44

4. ResultsWith det Hw2 (Preliminary)

Topology conservation seems perfect !

The overlap Dirac operatorWe 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.

4. Results

The condition number

The gain is about a factor 2-3.

4. Results

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)1.0 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)

quenched

The condition number

4. Results

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) 02/3 1.75 6.26(36) >46 0.1284(13) 0.080.0 5.2 6.16(19) >32 0.2325(17) 0.05

quenched 0 6.13 6.642 20 0.139(10) 0.03

With det Hw2 (Preliminary)

The localityFor

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

quenched

4. Results

We find New cooling method does work. In quenched study, the lattice spacing can be determined in a co

nventional 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.

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

5. Conclusion and Outlook

Topology dependence

Q dependence of the quark potential seems week

as we expected.

4. Results

size 1/ε β Initial Q Q stability plaquette r0/a rc/r0164 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)

topology conserving actions and the overlap dirac operator (hep-lat/0510116) hidenori fukaya yukawa...

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