Topological susceptibility and axial symm. at FTTopological susceptibility and axial symmetry
at finite temperature

Guido Cossu, KEK

高エネルギー加速器研究機構
Nemunosato Hotel – Kashikojima
December 4th 2011

Topological susceptibility and axial symm. at FT

Summary:✔ Motivation✔ Chiral phase transition at finite temperature and axial symmetry✔ Previous studies✔ Simulating dynamical overlap fermions✔ Topology fixing and friends✔ Methodology and results (quenched and Nf=2 )✔ Discussion and conclusions

People involved in the project
JLQCD group: S. Hashimoto, S. Aoki, T. Kaneko, H. Matsufuru,J. Noaki, E. Shintani
See for example POS(Lattice2010)174 (arXiv::1011.0257), article in prep. 2

Topological susceptibility and axial symm. at FTMotivation
Pattern of chiral symmetry breaking at low temperature QCD

SU (Nf )V £ SU (Nf )A £ U (1)V £ U (1)A ! SU (Nf )V £ U (1)VWell known facts●Axial anomaly – Non vanishing topological susceptibility●Mass splitting of the ´' with respect to the lighter mesons
What is the fate of the axial UA (1) symmetry
at finite temperature(T ≳ Tc)?
Dirac Overlap operator, retaining the maximal amount of chiral symmetryon the lattice is, theoretically, the best way to answer this question.
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Topological susceptibility and axial symm. at FT

Goal of the current project
Check restoration of axial UA (1) symmetry by measuring(spatial) meson correlators at finite temperature in full QCD with overlap
Degeneracy of correlators is the signal that we are looking for¾(14 ­ 12 )Chiral sym.

U (1)A´(i°5 ­ 12 )¼(i°5 ­ ¿ a )U (1)AChiral sym.

±(14 ­ ¿ a )As I will show in this talk,there are some issues to solve before attacking the real problem...

4Topological susceptibility and axial symm. at FT

Simulating dynamical overlap fermions
In order to avoid expensive tricks to handle the zero modes of theHermitian Wilson operator JLQCD simulations use (JLQCD 2006):● Iwasaki action (suppresses Wilson operator near zero modes)● Extra Wilson fermions and twisted mass ghosts to rule out the zeromodes
Topology is thus fixed throughout the HMC trajectory.
The effect of fixing topology is expected to be a Finite Size Effect(actually O(1/V) )·¸·µ¶¸
21Qc41 2ZQ = pexp ¡1¡+O;±:
222Ât V8V ÂtV2¼Ât V5

Topological susceptibility and axial symm. at FTFixing topology: how to deal with it at T=0
From the previous partition function we can extract the relation between
correlators at fixed θ and correlators at fixed Q
In particular for the topological susceptibility and using the Axial Ward
Identity we obtain a relation involving fermionic quantities:µ 2¶
1 Q
c4
disclim hmP (x)mP (0)iQ ´¡ Ât ¡+ O(e¡m¼ jxj )VV
2Ât Vjxj!largeP(x) is the flavor singlet pseudo scalar density operator
Aoki et al. PRD76,054508 (2007)
What is the effect of fixing Q at finite temperature?

6Topological susceptibility and axial symm. at FTResults✔ Simulation details✔ Eigenvalues distribution✔ Finite temperature quenched SU(3) at fixed topology✔ Meson correlators in two flavors QCD

BG/L

BG/Q(?)

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Topological susceptibility and axial symm. at FT

Simulation details

Pure gauge (163x6, 243x6):
Iwasaki action + top. fixing term


a(fm)

T (MeV)

T/Tc

2.35

0.132

249.1

0.86

2.40

0.123

268.1

0.93

2.43

0.117

280.9

0.97

2.44

0.115

285.7

0.992

2.445

0.114

288

1.0

2.45

0.1133

290.2

1.01

2.46

0.111

295.1

2.48

0.107

2.50
2.55

Two flavors QCD (163x8)
Iwasaki + Overlap + top. FixO(200) trj per T
am=0.05, 0.025, 0.01


a(fm)

T (MeV)

2.18

0.1438

171.5

0.95

2.200.1391

177.3

0.985

2.30

0.1183

208.5

1.15

1.02

2.40

0.1013

243.5

1.35

305.6

1.06

2.45

0.0940

262.4

1.45

0.104

316.2

1.10

0.094

347.6

1.20

T/Tc

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Topological susceptibility and axial symm. at FT

Eigenvalue distribution
Phase transition

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Topological susceptibility and axial symm. at FTTopological susceptibility in pure gauge theory - I1
disclim hmP (x)mP (0)iQ ´Vjxj!largeµ2Qc4¡ Ât ¡V2Ât V¶+ O(e¡m¼ jxj )(Spatial) Correlators are always approximated by the first 50
eigenvalues (enough to extract the x→ ∞ limit)●Pure gauge: double pole formula for disconnected diagram●Topological susceptibility estimated by a joint fit of connected

and disconnected contribution.

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Topological susceptibility and axial symm. at FTTopological susceptibility in pure gauge theory - II11

Topological susceptibility and axial symm. at FT

Full QCD – Eigenvalues
Effect of axial symmetry on
the Dirac sperctrum
Â¼¡± =

Z4m2
d¸½m (¸) 2¸ + m2

If chiral symmetry is
restored we can concludethat½m (¸)
lim ¸!0 lim m!0=0¸
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Topological susceptibility and axial symm. at FT

Full QCD – Meson correlators
Mass
Temperature

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Topological susceptibility and axial symm. at FT

Full QCD – Meson correlators
Mass
Temperature

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Topological susceptibility and axial symm. at FT

Full QCD – Meson correlators
Mass
Temperature

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Topological susceptibility and axial symm. at FT

Full QCD – Meson correlators
Mass
Temperature

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Topological susceptibility and axial symm. at FT

Full QCD – Meson correlators
MassTemperature17

Topological susceptibility and axial symm. at FT

Full QCD – Meson correlators
Mass
Temperature18

Topological susceptibility and axial symm. at FT

Full QCD – Meson correlators
Mass
Temperature

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Topological susceptibility and axial symm. at FT

Full QCD – Meson correlators
Mass
Temperature20

Topological susceptibility and axial symm. at FT

Full QCD – Meson correlators
Mass
Temperature21

Topological susceptibility and axial symm. at FT

Summary● Overlap fermions are the best choice to check axial anomaly at finitetemperature● Current machine and algorithms permit now realistic simulations...
● ...at the cost of fixing topology● We checked feasibility by test runs in pure gauge theory● In pure gauge systematic errors are under control.
● (almost) final results in Full QCD show restoration of UA (1) symmetry● We need more data on chiral limit and check approach to T (1-2 weeks)c22

Topological susceptibility and axial symm. at FT

Thanks for your attention

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LuxRay Artistic Rendering of Lowest Eigenmode

Topological susceptibility and axial symm. at FTBackup slides

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Lowest Eigenmode

Topological susceptibility in pure gauge theory - IIIRun IRun II25

Topological susceptibility and axial symm. at FTTopological susceptibility in pure gauge theory¯ = 2.50

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Topological susceptibility and axial symm. at FTTopological susceptibility in pure gauge theory - VII
Definition of the Inverse Participation Ratio½¸ (x) = Ã¸y (x)Ã¸ (x)X½¸ (x) = 1x

I¸ = VX½2¸ (x)x27

Topological susceptibility and axial symm. at FTTopological susceptibility in pure gauge theory - VII28

Topological susceptibility and axial symm. at FT29

Topological susceptibility and axial symm. at FT

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Topological susceptibility and axial symm. at FT

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Topological susceptibility and axial symm. at FT32