Transcript The fermion condensations and the ηmeson in θ vacuum

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)
1
Contents
1.
2.
3.
4.
5.
Introduction
The chiral symmetry and topology
Lattice simulations
Results
Summary and outlook
2
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;
CP-PACS 2002




Hadron spectrum
Matrix elements
Chiral transition
Quark gluon plasma
3
1. Introduction
But the lattice regularization spoils a lot of symmetries…




Translational symmetry
Lorentz invariance
Chiral symmetry or topology
Supersymmetry…
4
1. Introduction
Chiral symmetry
Ginsparg and Wilson, Phys.Rev.D25,2649(1982)
 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.
M.Luescher, Nucl.Phys.B538,515(1999)
 Additional Wilson fermion action with negative mass.
M.Luescher, Private communications
may solve the both theoretical and numerical problems.
5
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 0603:017,2006 .
6
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.
7
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 1280000 dimensions.
NOTE: each step is small ⇒ tendency of keeping topology.
8
2. The chiral symmetry and topology
Nielsen-Ninomiya theorem: Any local Dirac operator
satisfying
has unphysical poles (doublers).
Example - free fermion –
 Continuum
 Lattice
has unphysical poles at
 Wilson fermion
has no doubler.
.
Doublers are decoupled but spoils chiral symmetry.
Nielsen and Ninomiya, Nucl.Phys.B185,20(1981)
9
2. The chiral symmetry and topology

Eigenvalue distribution of Dirac operator
continuum
(massive)
1/a
0
-1/a
2/a
4/a
6/a
m
10
2. The chiral symmetry and topology

Eigenvalue distribution of Dirac operator
Wilson fermion
Naïve lattice
fermion
(massive)
1/a
16 lines
0
m
-1/a
2/a
4/a
6/a
• Doublers are massive.
• m is not well-defined.
• The index is not well-defined.
11
 The Ginsparg-Wilson fermion
The Neuberger’s overlap operator:
Phys.Lett.B417,141(‘98)
satisfying the Ginsparg-Wilson relation:
Phys.Rev.D25,2649(‘82)
realizes ‘modified’ exact chiral symmetry on the lattice;
the action is invariant under
NOTE
M.Luescher,Phys.Lett.B428,342(1998)
 Expansion in Wilson Dirac operator ⇒ No doubler.
 Fermion measure is not invariant;
⇒ chiral anomaly, index theorem
12
2. The chiral symmetry and topology

Eigenvalue distribution of Dirac operator
The overlap fermion
1/a
0
2/a
4/a
6/a
-1/a
• D is smooth except for
.
13
2. The chiral symmetry and topology

Eigenvalue distribution of Dirac operator
The overlap fermion
(massive)
1/a
0
m
2/a
4/a
6/a
-1/a
• Doublers are massive.
• m is well-defined.
14
2. The chiral symmetry and topology

Eigenvalue distribution of Dirac operator
The overlap fermion
1/a
0
-1/a
2/a
4/a
6/a
• Theoretically ill-defined.
• Large simulation cost.
15
2. The chiral symmetry and topology

The topology (index) changes
The complex modes make pairs
1/a
0
-1/a
Whenever Hw crosses zero,
topology changes.
2/a
The real modes6/a
4/a
are chiral
eigenstates.
16
 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.
17
2. The chiral
SG symmetry and topology

ε< 1/20
The topology
conserving gauge action
ε= 1.0
generates configurations satisfying Luescher’s
“admissibility” condition:
M.Luescher,Nucl.Phys.B568,162 (‘00)
NOTE:
Q=0
ε=∞
Q=1
 The effect of ε is O(a4) and the positivity is restored as
M.Creutz, Phys.Rev.D70,091501(‘04)
ε/a4 → ∞.
 |Hw| > 0 if ε ＜ 1/20.49, but it’ s too small…
If the barrier
is high enough,
Q may
fixed.ε.
Let’s be
try larger
P.Hernandez
et al, Nucl.Phys.B552,363
(1999))
18
 Admissibility in 2D QED
Topological charge is defined as
Without
condition ( ε＜ 2 ),
 If
gauge admissibility
fields are “admissible”
⇒ topological charge can jump； Q → Q±１.
⇒ topological charge is conserved !!
19
2. The chiral symmetry and topology
 The negative mass Wilson fermion
 would also suppress the topology changes.
 would not affect the low-energy physics.
20
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
21
size
124
1/ε
1.0
β
Δτ
Nmds acceptance
Plaquette
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)
 Topology2.4conserving
action (quenched)
0.01
40gauge93%
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
82%
0.57840(1)
 with
1/ε=
1.0, 2/3, 20
0.0 (=plaquette
action)
.
1.42 0.01
20
82%
0.59167(1)
 Algorithm:
The standard
HMC
method. 0.58428(2)
2/3 2.55 0.01
20
88%
20 4,204 .87%
0.59862(1)
 Lattice2.7
size 0.01
: 124,16
6.0 0.01
20
89%
0.59382(5)
 10.0
trajectory
= 20 - 40
molecular
dynamics
steps
6.13 0.01
40
88%
0.60711(4)
with
Δτ= 0.01
204
1.0 stepsize
1.3 0.01
20 - 0.02.
72%
0.57847(9)
1.42 0.01
20
74%
0.59165(1)
simulations
were done on0.58438(2)
the Alpha work
2/3 2.55 0.01 The20
82%
and SX-5 at RCNP.
2.7 0.01 station
20 at YITP82%
0.59865(1)
0.0
6.0 0.015
20
53%
0.59382(4)
22
6.13 0.01
20
83%
0.60716(3)
3. Lattice simulations
3. Lattice simulations
 Negative mass Wilson fermion
size
144
1/ε
β
Δτ
Nmds acceptance
Plaquette
1.0 0.75 0.01
15
72%
0.52260(2)
2/3
1.8 0.01
15
87%
0.52915(3)
With
0.0 s=0.6.
5.0 0.01
15
88%
0.55377(6)
1.0
0.8conserving
0.007
60
79% (1/ε=1,2/3,0)
0.53091(1)
Topology
gauge action
2/3 1.75 0.008
50
89%
0.52227(4)
Algorithm: HMC + pseudofermion
0.0
5.2 0.008
50
93%
0.57577(3)

4
16

 Lattice size : 144,164 .
 1 trajectory = 15 - 60 molecular dynamics steps
with stepsize Δτ= 0.007-0.01.
The simulations were done on the Alpha work
station at YITP and SX-5 at RCNP.
23
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/software/
24
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)
25
3. Lattice simulations
The
agreement
of Q
with cooling
and the index
of
 New
cooling
method
to measure
Q
overlap
D the
is roughly
(with only
20-80
We “cool”
configuration
smoothly
bysamples)
performing HMC
 ~with
90-95%
for 1/ε= increasing
1.0 and 2/3.
steps
exponentially
 ~ 60-70% for 1/ε=0.0 (plaquette action)
(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 .
26
4. Results
With det Hw2
quenched
 The static quark potential
In the following, we assume Q does not affect the
Wilson loops. ( initial Q=0 )
1. We measure the Wilson loops,
in
6 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
27
4. Results
 The static quark potential
In the following, we assume Q does not affect the
Wilson loops. ( initial Q=0 )
1. We measure the Wilson loops,
in
6 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
28
4. Results
 The static quark potential
size
124
164
1/ε
1.0
quenched
β samples
r0/a
rc/a
1.0
3800 3.257(30) 1.7081(50)
1.2
3800 4.555(73) 2.319(10)
1.3
3800 5.140(50) 2.710(14)
2/3 2.25 3800 3.498(24) 1.8304(60)
2.4
3800 4.386(53) 2.254(16)
2.55 3800 5.433(72) 2.809(18)
1.0 1.3
2300 5.240(96) 2.686(13)
1.42 2247 6.240(89) 3.270(26)
2/3 2.55 1950 5.290(69) 2.738(15)
2.7
2150 6.559(76) 3.382(22)
Continuum limit (Necco,Sommer ‘02)
a
~0.15fm
~0.11fm
~0.10fm
~0.14fm
~0.11fm
~0.09fm
~0.10fm
~0.08fm
~0.09fm
~0.08fm
rc/r0
0.5244(52)
0.5091(81)
0.5272(53)
0.5233(41)
0.5141(61)
0.5170(67)
0.5126(98)
0.5241(83)
0.5174(72)
0.5156(65)
0.5133(24)
Here we assume r0 ~ 0.5 fm.
29
4. Results
 The static quark potential
size
164
144
With det Hw2 (Preliminary)
1/ε
β samples
r0/a
rc/a
1.0 0.8
153
5.12(61) 2.473(51)
2/3 1.75
145
4.63(29) 2.307(60)
0
5.2
225
7.09(17) 3.462(55)
1.0 0.75
162
4.24(15) 2.240(37)
2/3 1.8
261
4.94(19) 2.361(26)
0
5.0
162
4.904(90) 2.691(42)
Continuum limit (Necco,Sommer ‘02)
a
~0.10fm
~0.11fm
~0.07fm
~0.12fm
~0.10fm
~0.10fm
rc/r0
0.483(56)
0.498(34)
0.489(13)
0.528(24)
0.478(19)
0.549(13)
0.5133(24)
30
4. Results
 Renormalization of the coupling
quenched
The renormalized coupling in Manton-scheme is defined
where
is the tadpole improved bare coupling:
where P is the plaquette expectation value.
R.K.Ellis,G.Martinelli, Nucl.Phys.B235,93(‘84)Erratum-ibid.B249,750(‘85)
31
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.
32
4. Results
 The stability of the topological charge
We measure Q using cooling per 20 trajectories
: auto correlation for the plaquette
M.Luescher, hep-lat/0409106 Appendix E.
: 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.
33
size
124
164
204
1/ε
1.0
2/3
0.0
1.0
2/3
0.0
1.0
2/3
0.0
1.0
2/3
0.0
1.0
2/3
0.0
1.0
2/3
0.0
1.0
2/3
0.0
β
1.0
2.25
5.8
1.2
2.4
5.9
1.3
2.55
6.0
1.3
2.55
6.0
1.42
2.7
6.13
1.3
2.55
6.0
1.42
2.7
6.13
r0/a
3.398(55)
3.555(39)
[3.668(12)]
4.464(65)
4.390(99)
[4.483(17)]
5.240(96)
5.290(69)
[5.368(22)]
5.240(96)
5.290(69)
[5.368(22)]
6.240(89)
6.559(76)
[6.642(-)]
5.240(96)
5.290(69)
[5.368(22)]
6.240(89)
6.559(76)
[6.642(-)]
Trj
18000
18000
18205
18000
18000
27116
18000
18000
27188
11600
12000
3500
5000
14000
5500
1240
1240
1600
7000
7800
1298
τplaq
2.91(33)
5.35(79)
30.2(6.6)
1.59(15)
2.62(23)
13.2(1.5)
1.091(70)
2.86(33)
15.7(3.0)
3.2(6)
6.4(5)
11.7(3.9)
2.6(4)
3.1(3)
12.4(3.3)
2.6(5)
3.4(7)
14.4(7.8)
3.8(8)
3.5(6)
9.3(2.8)
#Q Q stability
696
9
673
5
728
1
265
43
400
17
761
3
69
239
123
51
304
6
78
46
107
18
166
1.8
2
961
6
752
22
20
14
34
15
24
37
3
29
63
20
110
4
35
quenched
34
4. Results
With det Hw2
size
164
144
1/ε
1.0
2/3
0.0
1.0
2/3
0.0
β
0.8
1.75
5.2
0.75
1.8
5.0
r0/a
5.12(61)
4.63(29)
7.09(17)
4.24(15)
4.94(19)
4.904(90)
Trj
480
454
730
3500
5370
3120
τplaq
0.65(1)
1.8(5)
1.5(3)
5.1(8)
11(2)
21(6)
#Q Q stability
0
>693
0
>483
0
>146
0
>741
0
>251
0
>474
Topology conservation seems perfect !
35
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.
36
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 (<<0.1) are
suppressed.
plaquette
1/ε=2/3
1/ε=1
plaquette
with det Hw2
r0~6.5-7
37
4. Results
 The locality
quenched
For
should exponentially decay.
1/a~0.08fm
(with 4 samples),
no remarkable
improvement of
locality is seen…
＋ : beta = 1.42, 1/e=1.0
× : beta = 2.7, 1/e=2/3
＊ : beta = 6.13, 1/e=0.0
⇒ lower beta?
38
 How to sum up the different topological sectors
⇒ We need
.
39
 How to sum up the different topological sectors
 Formally,
 With an assumption,
The ratio can be given by the topological susceptibility,
V’
if it has small Q and V’ dependences.
 Parallel tempering + Fodor method may also be useful.
Z.Fodor et al. hep-lat/0510117 40
 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
41
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.
42
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.
43
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.
44
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…
45
(Pion mass)2 vs quark mass
6.
Nf=2
lattice
QCD
at
KEK
[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 16332 is underway.
 First result with exact chiral symmetric Dirac operator is
coming soon.
46
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
47
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 16332 is underway.
 First result with exact chiral symmetric Dirac operator is
coming soon.
48
4. Results
 Topology dependence
Q dependence of the quark potential seems week
as we expected.
size 1/ε
164 1.0
β
Initial Q Q stability plaquette
r0/a
rc/r0
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)
49
4. Results
 The condition number
size
204
164
1/ε
1.0
2/3
0.0
1.0
2/3
0.0
1.0
2/3
0.0
β
1.3
2.55
6.0
1.42
2.7
6.13
1.42
2.7
6.13
r0/a
5.240(96)
5.290(69)
5.368(22)
6.240(89)
6.559(76)
6.642(-)
6.240(89)
6.559(76)
6.642(-)
Q stability
34
24
3
63
110
35
961
752
20
quenched
1/κ
0.0148(14)
0.0101(08)
0.0059(34)
0.0282(21)
0.0251(19)
0.0126(15)
0.0367(21)
0.0320(19)
0.0232(17)
P(<0.1)
0.090(14)
0.145(12)
0.414(29)
0.031(10)
0.019(18)
0.084(14)
0.007(5)
0.020(8)
0.030(10)
50
4. Results
 The condition number
size
164
1/ε
β
r0/a
1.0 0.8 5.7(1.0)
2/3 1.75 6.26(36)
0.0 5.2 6.16(19)
quenched 0 6.13
6.642
With det Hw2 (Preliminary)
Q stability
hwmin
P(<0.1)
>43
>46
>32
20
0.1823(33)
0.1284(13)
0.2325(17)
0.139(10)
0
0.08
0.05
0.03
51