Analysis of topological structure of the QCD vacuum with overlap

Analysis of topological structure of the QCD vacuumwith overlap-Dirac operator eigenmode

T.Iritani

High Energy Accelerator Research Organization (KEK)

Jul 29, 2013

Coll. S. Hashimoto and G. Cossu

...1 Introduction

...2 Topological structure of QCD vacuum from overlap-Dirac eigenmodes

...3 Flux-tube Formation by Low-lying Dirac Eigenmodes

...4 Chiral Condensate in Flux-tube

...5 Summary

...1 Introduction




...5 Summary

Vacuum Structure of QCD

QCD vacuum is filled with interesting objects.

instanton/anti-instanton

⇒ chiral symmetry breaking

flux-tube

⇒ quark confinement

Figs (left) JLQCD Coll. ’12 (right) www.physics.adelaide.edu.au/theory/staff/leinweber/VisualQCD/Nobel/

1 / 15

Overlap-Dirac Operator

Dov = m0 [1 + γ5sgn HW (−m0)]

HW (−m0): hermitian Wilson-Dirac operator with a mass −m0 (Neuberger ’98)

overlap-Dirac eigenmode is an ideal probe to study QCD vacuum

exact chiral symmetry on the lattice

Banks-Casher relation— chiral symmetry breaking

〈q̄q〉 = −πρ(0)

index theorem — topology of QCD

12tr [γ5Dov] = nL − nR

0

0.001

0.002

0.003

0.004

0.005

0 0.01 0.02 0.03 0.04 0.05 0.06π

ρ Q(λ

)

λ

latticeChPT LO

ChPT NLO

Dirac spectral density ρ(λ)JLQCD Coll. ’10

2 / 15

Contents

...1 topological structure of QCD vacuum from overlap-Dirac eigenmodes

...2 flux-tube formation from overlap-Dirac eigenmodes

...3 chiral condensate in flux-tube

...1 Introduction




...5 Summary

Dirac Eigenmode Decomposition of Field Strength

Dirac eigenmodes ψλ(x) ⇒ Field strength tensor Fµν

Gattringer ’02

[ /D(x)]2 =∑

µ

D2µ(x) +

∑µ<ν

γµγνFµν(x)

∴ Fµν(x) = −14tr

[γµγν /D2(x)

]∝

∑λ

λ2fµν(x)λ

with Dirac-mode components fµν(x)λ

fµν(x)λ ≡ ψ†λ(x)γµγνψλ(x)

references Gattringer ’02, Ilgenfritz et al. ’07, ’08.In this talk, we usedynamical overlap-fermion configurations by JLQCD Coll.β = 2.3 (Iwasaki action), a ∼ 0.11 fm, Volume 163 × 48,overlap-Dirac operator Nf = 2 + 1, mud = 0.015, ms = 0.080

3 / 15

Duality of Field Strength Tensor by Overlap-eigenmodelowest eigenmode components of fµν

f12(x)0

20 22 24 26 28 30 32 34 0 2 4 6 8 10 12 14 16-3E-04

-2E-04

-1E-04

0E+00

1E-04

f012(x) 0 around (8, 15, 7, 27)

t x

⇒ negative peak

f34(x)0

20 22 24 26 28 30 32 34 0 2 4 6 8 10 12 14 16-1E-04

0E+00

1E-04

2E-04

3E-04

f034(x) 0 around (8, 15, 7, 27)

t x

⇒ positive peak

at this point, an anti-self-dual lump exists

f12(x)0 ' −f34(x)0

4 / 15

Action and Topological Charge from Dirac Eigenmodesaction density

ρ(N)(x) =N∑i,j

λ2iλ

2j

2fa

µν(x)ifaµν(x)j

topological charge density

q(N)(x) =N∑i,j

λ2iλ

2j

2fa

µν(x)if̃aµν(x)j

Figure: action and topological charge densities of lowest eigenstate

20 22 24 26 28 30 32 34 0 2 4 6 8 10 12 14 160E+00

2E-16

4E-16

6E-16

8E-16

Action Density #0 around (8, 15, 7, 27)

t x

20 22 24 26 28 30 32 34 0 2 4 6 8 10 12 14 16-8E-16

-6E-16

-4E-16

-2E-16

0E+00

Topological Charge Density #0 around (8, 15, 7, 27)

t x

there is an “anti-instanton” at this point5 / 15

...1 Introduction




...5 Summary

Flux-tube Formation Between Quarksaction density is “expelled” between quarks

⇒ formation of flux-tube ⇒ confinement potential

we discuss flux-tube structure from low-lying eigenmodes

flux-tube betweenquark and antiquark

Bali-Schilling-Schlichter ’95

Y-type flux-tube for 3Q-system

Ichie et al. ’036 / 15

Measurement of Flux-tube Structuredifference of action density ρ(x) with or without Wilson loop W (R, T )

〈ρ(x)〉W ≡ 〈ρ(x)W (R, T )〉〈W (R, T )〉

− 〈ρ〉 < 0

N.B. action density is lowered in the flux-tube

XY

Tim

e

Wilson Loop

action density

R7 / 15

Flux-tube Formation by Low-lying Eigenmodesρ(N)(x): low-mode truncated action density

〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉

〈W (R, T )〉− 〈ρ〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

8

inter-quark separation R = 10

XY

(−) 〈ρ(x)〉(N)W N = 100 eigenmodes

low-modes reallycontribute toflux-tube

1002, 359, 296

∼ 0.00004

λmax ∼ 300 MeV

8 / 15


〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉

〈W (R, T )〉− 〈ρ〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

8


XY



1002, 359, 296

∼ 0.00004

λmax ∼ 300 MeV

8 / 15


〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉

〈W (R, T )〉− 〈ρ〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

8


XY



1002, 359, 296

∼ 0.00004

λmax ∼ 300 MeV

8 / 15


〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉

〈W (R, T )〉− 〈ρ〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

8


XY



1002, 359, 296

∼ 0.00004

λmax ∼ 300 MeV

8 / 15


〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉

〈W (R, T )〉− 〈ρ〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

8


XY



1002, 359, 296

∼ 0.00004

λmax ∼ 300 MeV

8 / 15


〈ρ(x)〉(N)W ≡ 〈ρ(N)(x)W (R, T )〉

〈W (R, T )〉− 〈ρ〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

8


XY



1002, 359, 296

∼ 0.00004

λmax ∼ 300 MeV

8 / 15

Cross Section of Flux-tube

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

8


XY

-4e-10

-2e-10

0e+00

-8 -6 -4 -2 0 2 4 6 8

Y

σ = 2.42 ± 0.08R = 8

gaussian formρ(y) ∝ e−y2/σ2

thickness of flux∼ 0.6 fm

9 / 15

About Dirac Eigenmodes and Confinementconfinement remains without low or intermediate or higher eigenmodes

Refs. Gongyo-TI-Suganuma ’12, TI-Suganuma ’13

⇒ “seeds” of confinement seem to be widely distributed in eigenmodes

removing Dirac eigenmodes

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

ρ (λ

) V

[a

]

λ [a-1]

interquark potential

0.5

1

1.5

0 1 2 3V

QQ–

(R

) [a

-1]

R [a]


Figs. Gongyo-TI-Suganuma ’12

flux-tube formation by low-lying modes

⇒ direct evidence of the seed of confinement in the low-lying modes6= low-lying modes are essential for confinement

10 / 15





0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

ρ (λ

) V

[a

]

λ [a-1]


0.5

1

1.5

0 1 2 3V

QQ–

(R

) [a

-1]

R [a]

no cut

Dirac-mode cut




10 / 15





0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

ρ (λ

) V

[a

]

λ [a-1]


0.5

1

1.5

0 1 2 3V

QQ–

(R

) [a

-1]

R [a]

no cut

Dirac-mode cut




10 / 15





0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

ρ (λ

) V

[a

]

λ [a-1]


0.5

1

1.5

0 1 2 3V

QQ–

(R

) [a

-1]

R [a]

no cut

Dirac-mode cut




10 / 15





0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

ρ (λ

) V

[a

]

λ [a-1]


0.5

1

1.5

0 1 2 3V

QQ–

(R

) [a

-1]

R [a]

no cut

Dirac-mode cut




10 / 15





0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

ρ (λ

) V

[a

]

λ [a-1]


0.5

1

1.5

0 1 2 3V

QQ–

(R

) [a

-1]

R [a]

no cut

Dirac-mode cut




10 / 15





0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

ρ (λ

) V

[a

]

λ [a-1]


0.5

1

1.5

0 1 2 3V

QQ–

(R

) [a

-1]

R [a]

no cut

Dirac-mode cut




10 / 15





0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

ρ (λ

) V

[a

]

λ [a-1]


0.5

1

1.5

0 1 2 3V

QQ–

(R

) [a

-1]

R [a]

no cut

Dirac-mode cut




10 / 15





0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5

ρ (λ

) V

[a

]

λ [a-1]


0.5

1

1.5

0 1 2 3V

QQ–

(R

) [a

-1]

R [a]

no cut

Dirac-mode cut




10 / 15

...1 Introduction




...5 Summary

Dirac Eigenmodes and Chiral Condensatechiral condensate 〈q̄q〉

〈q̄q〉 = −Tr1

/D +mq= − 1

V

∑λ

1λ+mq

cf. Banks-Casher relation 〈q̄q〉 = −πρ(0)

“local chiral condensate” q̄q(x) is given by

q̄q(x) = −∑

λ

ψ†λ(x)ψλ(x)λ+mq

with Dirac eigenmodes ψλ(x) 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16-0.005

-0.004

-0.003

-0.002

-0.001

0

sample of a local condensate11 / 15

Chiral Condensate in Flux-tubedifference of chiral condensate q̄q(x) with or without Wilson loop W (R, T )

〈q̄q(x)〉W ≡ 〈q̄q(x)W (R, T )〉〈W (R, T )〉

− 〈q̄q〉

XY

Wilson Loop

local chiral condensate

R12 / 15

Reduction of Chiral Condensate in Fluxdifference of “truncated local chiral condensate”

〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉

〈W (R, T )〉− 〈q̄q〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

80.0000

0.0002

0.0004

0.0006 R = 10

XY

mq = 0.015 using N = 100 eigenmodes

〈q̄q(x)〉(N)W > 0

partially restored|〈q̄q〉|in flux < |〈q̄q〉|“Bag-model” like

cf. chiral condensate in color-electro/magnetic fields (Suganuma-Tatsumi ’93)

13 / 15


〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉

〈W (R, T )〉− 〈q̄q〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

80.0000

0.0002

0.0004

0.0006 R = 9

XY


〈q̄q(x)〉(N)W > 0



13 / 15


〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉

〈W (R, T )〉− 〈q̄q〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

80.0000

0.0002

0.0004

0.0006 R = 8

XY


〈q̄q(x)〉(N)W > 0



13 / 15


〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉

〈W (R, T )〉− 〈q̄q〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

80.0000

0.0002

0.0004

0.0006 R = 7

XY


〈q̄q(x)〉(N)W > 0



13 / 15


〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉

〈W (R, T )〉− 〈q̄q〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

80.0000

0.0002

0.0004

0.0006 R = 6

XY


〈q̄q(x)〉(N)W > 0



13 / 15


〈q̄q(x)〉(N)W ≡ 〈q̄q(N)(x)W (R, T )〉

〈W (R, T )〉− 〈q̄q〉(N)

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

80.0000

0.0002

0.0004

0.0006 R = 5

XY


〈q̄q(x)〉(N)W > 0



13 / 15

Reduction Ratio of Chiral Condensate in Flux-tube

r(x) ≡ 〈q̄q(subt)(x)W (R, T )〉〈q̄q(subt)〉〈W (R, T )〉

< 1

-8-6

-4-2

0 2

4 6

8

-8-6

-4-2

0 2

4 6

80.0000

0.0002

0.0004

0.0006 R = 8

XY

0.7

0.8

0.9

1.0

-8 -6 -4 -2 0 2 4 6 8

r (x), Y-axis with R = 8

at center of flux

reduction of |〈q̄q〉|⇒ about 20 %

cf. subtracted condensate 〈q̄q〉(N) = 〈q̄q(subt)〉 + c(N)1 mq/a2 + c

(N)2 m3

q

14 / 15

...1 Introduction




...5 Summary

Summary

we discuss vacuum structure of QCD using overlap-Dirac eigenmodes

low-lying overlap-Dirac eigenmodes⇒ show “instanton”-like behavior

I (anti-)self-dual field strength fµν ' (−)f̃µν

I (anti-)self-dual lump of action density / topological charge density

flux-tube formation by low-lying Dirac eigenmodes

⇒ low-lying eigenmodes contribute to the formation of flux-tubei.e., confinement

chiral condensation in the flux-tube

⇒ chiral symmetry is partially restored in the flux-tubereduction of |〈q̄q〉| is about 20% at the center of flux

outlooks

chiral condensate in 3Q-system, quark-number densities in theflux-tube, light-quark content in quarkonia, topological susceptibilityfrom eigenmodes, and so on

15 / 15

...6 Appendix

Appendix

Lattice Setup

gauge configurationsI JLQCD dynamical overlap simulation Nf = 2 + 1I 163 × 48, β = 2.3, mud = 0.015, ms = 0.080, Q = 0I a−1 = 1.759(10) GeV (mπ ∼ 300 MeV)I 100 low-lying Dirac eigenmodes (λUV ∼ 400 MeV)I 50 configurations

Wilson loop W (R, T ) measurementI APE smearing for spatial link-variable Ui — 16 sweepsI T = 4 — ground-state component is dominantI measurement of action density/local chiral condensate at T = 2

Subtracted Chiral Condensate

〈q̄q〉(N) = 〈q̄q(subt)〉 + c(N)1

mq

a2+ c

(N)2 m3

q

0

0.001

0.002

0.003

0.004

0.005

0 0.05 0.1 0.15 0.2

- 〈 q

bar q

〉(N

)

mq

N = 30

N = 50

0

0.001

0.002

0.003

0.004

0.005

0 0.05 0.1 0.15 0.2- 〈 q

bar q〉(s

ubt)

mq

N = 30

N = 50

Figure: (a) 〈q̄q〉(N) (b) 〈q̄q〉(subt)

ref. J. Noaki et al., Phys. Rev. D81, 034502 (2010).

Documents

Analysis of topological structure of the QCD vacuum with overlap