Finite temperature study of axial symmetry on the lattice

Finite temperature study ofaxial symmetry on the lattice

Guido Cossu
High Energy Accelerator Research Organization – KEK
高エネルギー加速器研究機構

Quarks to Universe in Computational Science (QUCS 2012)
Nara, December 14th 2012

Finite temperature study of axial symmetry on the lattice

Summary✔ Introduction – chiral symmetry at finite temperature✔ Simulations with dynamical overlap fermions✔ Fixing topology✔ Results:✔ (test case) Pure gauge✔ Nf=2 case
✔ Other studies✔ Discussion and conclusions
People involved in the collaboration:
JLQCD group: H. Fukaya, S. Hashimoto, S. Aoki, T. Kaneko,
H. Matsufuru, J. Noaki
See previous Lattice proceedings (10-11-12), article in prep.

Finite temperature study of axial symmetry on the lattice

Introduction – chiral symmetry
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)VSymmetry of the Lagrangian

Symmetries in the real world (Nf=3 ) at zero temperature● U(1)V the baryon number conservation● SU(3)
V intact (softly broken by quark masses) – 8 Goldstone bosons (GB)
● SU(3)
A is broken spontaneously by the non zero e.v. of the quark condensate● No opposite parity GB, U(1)A is broken, but no 9th GB is found in nature.
Axial symmetry is not a symmetry of the quantum theory ('t Hooft - instantons)@¹ j5¹ (x) = 2iNf q(x) topological charge density
Witten-Veneziano: mass splitting of the ´'(958) from topological charge at large N.

Finite temperature study of axial symmetry on the lattice

Introduction – finite temperature
At finite temperature,in the chiral limit mq → 0, chiral symmetry is restored● Phase transition at N =2f● Crossover with 2+1 flavors
What is the fate of the axial U(1)A symmetry
at finite temperature T ≳ Tc ?
Complete restoration is not possible since it is an anomaly effect.
Exact restoration is expected only at infinite T (see instanton-gas models)At most we can observe strong suppression
On the lattice the overlap Dirac operator is the best way to answer this questions
since it preserves the maximal amount of chiral symmetry.
An operator satisfying the Ginsparg-Wilson relation has several nice properties e.g.
● exact relations between eigenmodes (EM) and topological charge● exact relation among hermitian-operator EM and the non-hermitian ones
● ...

Finite temperature study of axial symmetry on the lattice

Project intent
Check the effective restoration of axial U(1)A symmetry by measuring(spatial) meson correlators at finite temperature in full QCD with the Overlapoperator
Degeneracy of the correlators is the signal that we are looking for¾(14 ­ 12 )U (1)A´(i°5 ­ 12 )¼(i°5 ­ ¿ a )U (1)A±(14 ­ ¿ a )Dirac operator eigenvalue density is also a relevant observable for chiral symmetryFirst of all there are some issues to solve before dealing with the real problem...

Finite temperature study of axial symmetry on the lattice

Simulations with Overlap fermions
The sign function in the overlap operator gives
a delta in the force when HW modes cross the boundary(i.e. topology changes).
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) ), next slides

Finite temperature study of axial symmetry on the lattice

Fixing Topology: zero temperaturePartition function at fixed topology

1ZQ =2¼Z ¼dµ exp(¡V F (µ))¡¼F (µ) ´ E(µ) ¡ iµQ=V1X
c2n 2nÂt 2
where the ground state
4E(µ)=µ=µ+O(µ)
0energy can be expanded(2n)!
2n=1(T=0)Q(1 + O(± 2 ))Using saddle point expansion around µc = iÂt V
one obtains the Gaussian distribution·¸·µ¶¸
21Q
c4
1 2ZQ = pexp ¡1¡+O;±:
222Ât V8V ÂtV2¼Ât V

Finite temperature study of axial symmetry on the lattice

Fixing Topology
From the previous partition function we canextract the relation between correlators at fixed µ andcorrelators 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 )VV2Ât 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?

Finite temperature study of axial symmetry on the lattice

Results✔ Simulation details✔ Finite temperature quenched SU(3) at fixed topology (TEST CASE)✔ Eigenvalues density distribution✔ Topological susceptibility✔ Finite temperature two flavors QCD at fixed topology✔ Eigenvalues density distribution✔ Meson correlators

BG/L
Hitachi SR16K

Finite temperature study of axial symmetry on the lattice

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

a(fm)

T (MeV)

T/Tc

Two flavors QCD (163x8)
Iwasaki + Overlap + top. Fix
O(300) trajectories per T
am=0.05, 0.025, 0.01

2.35

0.132

249.1

0.86

2.40

0.123

268.1

0.93



a(fm)

T (MeV)

2.43

0.117

280.9

0.97

2.18

0.1438

171.5

0.95

2.44

0.115

285.7

0.992

2.20

0.1391

177.3

0.985

2.445

0.114

288

1.0

2.45

0.1133

290.2

1.01

2.25

0.12818

192.2

1.06

2.46

0.111

295.1

1.02

2.30

0.1183

208.5

1.15

2.48

0.107

305.6

1.06

2.40

0.1013

243.5

1.35

2.50

0.104

316.2

1.10

2.45

0.0940

262.4

1.45

2.55

0.094347.6

1.20

T/Tc

Pion mass: ~290 MeV @ am=0.015 , ¯=2.30
Tc was conventionally fixed to 180, not relevant for
the results (but supported by Borsanyi et al. results)

Finite temperature study of axial symmetry on the lattice

Eigenvalues distribution SU(3)Phase transition

Finite temperature study of axial symmetry on the lattice

Topological susceptibility

lim

jxj!large

hmP (x)mP (0)idisc´Q1
Vµ2

Qc4¡ Ât ¡V2Ât V¶+ O(e¡m¼ jxj )(Spatial) Correlators are always approximated by the first 50
eigenvaluesPure gauge: double pole formula for disconnected diagramQ=0, assume c4 term is negligible, then check consistencyTopological susceptibility estimated by a joint fit of connected anddisconnected contribution to maximize info from dataFinite temperature study of axial symmetry on the lattice

Topological susceptibility

Finite temperature study of axial symmetry on the lattice

Eigenvalues distribution - QCD N f =2

Effect of axial symmetryon the Dirac spectrum¼¡±

Â

=Z 104m2d¸½m (¸) 2¸ + m2

If axial symmetry is
restored we can obtain
constraints
onthespectral density
lim ¸!0 lim m!0½m (¸)=02¸

Ref: S. Aoki, H. Fukaya, Y. Taniguchi
arXiv:1209.2061

Finite temperature study of axial symmetry on the lattice

Eigenvalues distribution – Approximation

Decreasing Zolotarev
poles in the approx.
of the sign function
At Npoles = 5 morenear zero modes
appearFinite temperature study of axial symmetry on the lattice

Other analyses

Vranas 2000
DWFNo restoration

Bazavov et al. 2011(HotQCD) DWF
Low modes

Ohno et al. 2011 – HISQ (2+1) tails

Finite temperature study of axial symmetry on the lattice

Meson Correlators
MassTemperature

Finite temperature study of axial symmetry on the lattice

Meson Correlators
Mass
Temperature

Finite temperature study of axial symmetry on the lattice

Meson Correlators
MassTemperature

Finite temperature study of axial symmetry on the lattice

Meson Correlators
MassTemperature

Finite temperature study of axial symmetry on the lattice

Meson Correlators
MassTemperature

Finite temperature study of axial symmetry on the lattice

Meson Correlators
MassTemperature

Finite temperature study of axial symmetry on the lattice

Meson Correlators
MassTemperature

Finite temperature study of axial symmetry on the lattice

Meson Correlators
MassTemperature

Finite temperature study of axial symmetry on the lattice

Meson Correlators
Mass
Temperature

Finite temperature study of axial symmetry on the lattice

Meson Correlators
Mass
Temperature

Finite temperature study of axial symmetry on the lattice

Meson Correlators
Mass
TemperatureFinite temperature study of axial symmetry on the lattice

Conclusions
With overlap fermions we have a clear theoretical setup
for the analysis of spectral density and control on chirality violation terms.
● Realistic simulations are possible, but topology must be fixed● A check of systematics due to topology fixing at finite temperature is necessary(finite volume corrections expected)● Pure gauge test results show that we can control these errors as in the previous
T=0 case.
Finite volume effects are small in the SU(3) case
●Full QCD spectrum shows a gap at high temperature even at pion masses ~250
MeV
● Statistics: high at T>200 MeV● Finite volume effects (beside topology fixing): to be checked● Correlators show degeneracy of all channels when mass is decreased● Results support effective restoration of U(1)A symmetry●

Finite temperature study of axial symmetry on the latticeThanks for your attention

LuxRay Artistic Rendering of Lowest EigenmodeFinite temperature study of axial symmetry on the lattice

Backup slides

Finite temperature study of axial symmetry on the lattice

Meson Correlators
Mass
NB Volume effects and statistics

Temperature

Finite temperature study of axial symmetry on the latticeo mne gi E t s e woL

Pure gauge theory