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.
0 2/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
Eigenvalue distribution of Dirac operators
2. The overlap fermion and topology
1/a
-1/a
naïve fermion
16 lines
0 2/a 4/a 6/a
(massive)
m
• Doublers are massive.• m is not well-defined.
Eigenvalue distribution of Dirac operators
2. The overlap fermion and topology
1/a
-1/a
0 2/a 4/a 6/a
The overlap fermion
• D is smooth except for .
Eigenvalue distribution of Dirac operators
2. The overlap fermion and topology
1/a
-1/a
0 2/a 4/a 6/a
The overlap fermion(massive)
m
• Doublers are massive.• m is well-defined.
Eigenvalue distribution of Dirac operators
2. The overlap fermion and topology
1/a
-1/a
0 2/a 4/a 6/a
The overlap fermion
• Topology boundary.• Locality may be lost.• Large simulation cost.
The topology (index) changes
2. The overlap fermion and topology
1/a
-1/a
0 2/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)
2. The overlap fermion and topology
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…
2. The overlap fermion and topology
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…
2. The overlap fermion and topology
How to sum up the different topological sectors
2. The overlap fermion and topology
⇒ 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.
2. The overlap fermion and topology
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.
3. Lattice simulations
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.
3. Lattice simulations
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.
3. Lattice simulations
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 .
3. Lattice simulations
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) 0
2/3 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 dependenceQ 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)