Axial symmetry at finite
temperature and Diracoperator eigenmodes
Guido Cossu
High Energy Accelerator Organization (KEK)
Joint Institute for Computational Fundamental Science (JICFuS)YITP HHIQCD2015, Kyoto
March 5th 2015

Finitetemperature
Axial symm.
Introduction

Literature

Methods&Results

OutlineFinalthoughts

JLQCD collaborators
Shoji Hashimoto
Jun-ichi Noaki
Takashi Kaneko

KEK
Hidenori Fukaya
Akio Tomiya

Osaka Un.

Finitetemperature
Axial symm.
Introduction

Literature

Methods&Results

Finalthoughts

Low temperature – symmetries

Dirac operator eigenmodes

T<Tc

Chiralcondensate

Near zero modes density

Instantons

Zero modes

High temperature – symmetries

Current knowledge
Restoration of chiralsymmetry at 𝑇𝑐

No
condensate

Restoration at T → ∞Axial
symmetry

T>Tc

High temperature – symmetries

Nocondensate

𝑇 ≳ 𝑇𝑐 ?
T>Tc

Axial
symmetry?

Finitetemperature
Axial symm.
Introduction

Literature

Methods&Results

Finalthoughts

Recent literature - I
G. Cossu et al. (2013) for JLQCD

Disconnected meson diagrams
vanish at temperatures above Tc

Meson spatial correlatorsRelated: Gap in the Dirac spectrum
Aoki, Fukaya, Taniguchi (2012)

Analytic calculation (Overlap)
Dirac spectrum 𝜌(𝜆)~𝑐𝜆3
Implies U(1)A anomaly invisible

RestoredRecent literature - II
G. Cossu et al. (2013) for JLQCD

Disconnected meson diagrams
vanish at temperatures above Tc
Related: Gap in the Dirac spectrum
Aoki, Fukaya, Taniguchi (2012)

Analytic calculation (Overlap)
Dirac spectrum 𝜌(𝜆)~𝑐𝜆3
Implies U(1)A anomaly invisible

Dirac spectrum

Recent literature - IIIBazavov et al. (2012-13)
Domain wall, several volumes
Dirac spectrum, susceptibilities
NOT restored

Ohno et al., Sharma et al. (2012-15)
Overlap on HISQ configurations
Dirac spectrum
NOT restored
Brandt et al. (2013)
Wilson improved fermions
Screening masses
NOT restored

Our previous study
Exact chiral symmetry (Overlap)
topology fixed
Only 163x8 volume
Mass dependence
No continuum limit

Finitetemperature
Axial symm.
Introduction

Literature

Methods&Results

Finalthoughts

Chiral symmetry on the lattice
Nielsen-Ninomiya no-go theorem:
chiral symmetry implies unwanted doublers
The Ginsparg-Wilson relation (1982)

Generalized Domain WallPlay with the sign function
Möbius Kernel

Function approximation
Transfer matrix in 5D

 Hyperbolic tangent
 Rational approximation

Reduced residual massb=2 c=1 Scaled Shamir, mres ~ 10-4

Status of simulations
Symanzik + smeared 2-flavors DWF
Multipurpose code, HMC & measurements
Available on request, soon online
Optimized for BlueGene/Q
Webpage: http://suchix.kek.jp/guido_cossu/

Collected data

Full analysis in progress 2 lattice volumes
 3 masses
 5 temperatures
 Topology changes
 Nt=8, Nt=12 (finer lattice)Phase transition
Today:
T ~ 184, 200 MeV (red arrows)
Phase transition at ~180 MeV
2 volumes
Mass dependence
Nt=12 running now.
Analysis almost done

DeltaTheoretically clean: zero if axial symmetry is restored
Veeery delicate measurement
Talk: breakdown of the signal sources (physics/artifacts)
First: integrating correlators is bad, so
Stochastic measurements of Dirac traces
Eigenmode decompositionSource of the signal
Discrete spectral sum
Zero modes

Peaks dominate the signal76%
Fluctuations of3 orders of magnitude

One mode signal - Topology correlation
Mild correlation

Tension with spectral sum
expectations

Two sources
 GW violations
 𝐹 𝐹 estimate
Q=0 near zero modes

Temperature dependence
𝑇 < 𝑇𝑐Broad picture arisingat this stage: Just above the phasetransition zero modes
dominate
 Then they are strongly
suppressed and thesignal goes down

Let’s increase volume – m=0.01
Zero mode
contribution
suppressed ~1/V
As expected fromspectral sum
Bulk contribution
increases
Decrease the mass?

Let’s increase volume – m=0.005
Zero mode
contribution
suppressed ~1/V
As expected fromspectral sumBulk contribution
increases

Continuum limit
Nt from 8 to 12
No big news, violations
reducedT ~ 200 MeV

Volume&mass dependence
Zero modes
contribution
vanishes

Conclusion: signal from thebulk part, near zero modes
Let’s cut all configurations
with Q>0 (naïve cut)
Signal constant with the massIs everything all right? – I
From the Ginsparg-Wilson relation we can measure theamount of violation for each mode, 𝒈𝒏𝒏

Is everything all right? – II

Lowest modes show violations of GW by 1
order of magnitude bigger than the averageIs everything all right? – III
Exact result for susceptibility

GW violation terms

Is everything all right? – IVReweight it! (DWF to Overlap)
BeforeAfter200 MeV

76%

85%

Temperature and mass dependence
T ~ 184 MeV Just above Tc

Quark mass 10 times smaller

Agrees withOverlap result~3 orders ofmagnitude smaller
signalZero in chiral limit?

Quark mass dependenceU(1)Asymmetry
Finitetemperature

Literature

Methods&Results

Finalthoughts

Instanton gas – hints?
Results not yet conclusive(analysis running right now)
If the large volume signal isnot coming from lattice
artifacts
Near zero modes are
responsible for breaking U(1)What are they?
Poisson distributed?

Fun with 3D – put your glasses on
Norminstanton

anti-instanton

Eigenvalue

Chirality

IPR ~34

Fun with 3D – put your glasses on
Norm
Eigenvalue

Chirality

IPR ~34

Fun with 3D – put your glasses on
Norm
Eigenvalue

Chirality

IPR ~5

Localized modes

Summary – one more slide…
DWF volume &mass
dependence
suggests thatnear zero modes
are the source ofU(1) breaking

Latticeartifacts
can spoil
the signal

DWF lowest
modes look likean instanton
weaklyinteracting gas
Exact chiralsymmetry
results differ
from DWF

Are we finished?
The talk is over the work is not! (but almost there)
Some collected data yet to analyze
 Reweighting
 Continuum limit
Lattice artifacts?
 Chiral limit

Gas of instanton pairs, dyons?
Correlation with Polyakov loop?
U(1) restoration above criticaltemperature
is still an open question.

Axial symmetry at finite
temperature and Dirac
operator eigenmodes
Guido Cossu
High Energy Accelerator Organization (KEK)
Joint Institute for Computational Fundamental Science (JICFuS)YITP HHIQCD2015, KyotoMarch 5th 2015

Backupslides

GW violations

Many configurations violate GW

DW – OV eigenvalue mismatch

Susceptibility scales with volume

Let’s increase volume
Zero mode contribution ~1/V-Bulk contribution increases