Tsukuba Global Science Week (TGSW), 2021/9/11, online (Tsukuba) Ising Model
CriticalPoint

gasliquidsolid

CriticalPoint

1st tr.

Temperature Waterpressure

These CPs belong to the same universality class (𝑍2 ).
Common critical exponents.
ex.

 Building blocks of matter
nucleusnucleonquark Building blocks of matter
nucleusnucleonquark

 Quantum chromodynamics (QCD)=Fundamental theory of strong interaction

gluonsquarksgluonsvacuumbaryonmeson

vacuumbaryonmeson

temperaturevacuum

temperaturebaryonmeson

quark-gluon plasma

Early UniverseTemperature

Quark-Gluon Plasma

Critical
Point
HadronPhase

Color SC

Chemical PotentialTemperature

Quark-Gluon Plasma

Critical
Point
HadronPhase

Another CP?
MK+ (2002)

Color SC

Chemical PotentialTemperature

Quark-Gluon Plasma

Critical
Point
HadronPhase

Another CP?
MK+ (2002)

Color SC

Chemical Potential Columbia plot
= order of phase tr. at 𝜇𝑞 = 0

 Example
Phase diagram inheavy-quark region

Critical
Point

Various orders of phase transition with variation of 𝑚𝑞 .

Real Experiment

Virtual Experiment

Relativistic Heavy-Ion
Collisions

Lattice QCD
Numerical SimulationsColliding two heavy nuclei inaccelerators
RHIC (USA), LHC (EU), etc.

First-principle simulations
on supercomputersHadron Spectroscopy

PACS-CS,
PRD79 (’09)

Thermodynamics
Unique tool to perform
quantitative analyses of
non-perturbative QCD aspects

LHC – Large Hadron Collider

Proton-proton

protonHeavy Ion Collisions

protonElementary processes
new particle searchproperties of particles
LHC – Large Hadron Collider

Thermal Medium
hot & dense medium
phase transitions

temperature

LHCfuture facilities
ex. J-PARC @ Tokaibaryon chemical potentialtemperatureRHIC-BES
LHC

Phase I
2010~2015
Phase II
2019~2021

GSI-FAIR
future facilities
2026~?
ex. J-PARC @ Tokai

J-PARC-HI
2028~??

baryon chemical potentialtemperature

RHIC-BES
LHC

Phase I
2010~2015
Phase II
2019~2021

GSI-FAIR
future facilities
2026~?
ex. J-PARC @ Tokai

J-PARC-HI
2028~??

baryon chemical potential

Real Exp.

Virtual Exp.

Relativistic
Heavy-Ion Collisions

Lattice QCD
Numerical
Simulations

Real experiments

Virtual, but unphysical params

Various density

Small baryon density only

Dynamical evolution

Ideal thermal system

Final-state observables only

Limited observables

Complementary use of both exps. is important!

Observables in equilibrium are fluctuating!

varianceVBinder Cumulant

Cumulants

 skewness

 kurtosis

averagevariance

NOTE• Gauss distribution:• Poisson distribution:
Review: Asakawa, MK, PPNP 90 (2016)

•••𝑃(𝑀) : probability distr.
𝑉(𝑀) : effective potential
𝑀 : order parameter

1st tr.

Ising Model• Sign of 𝑁 3 𝑐 is flipped at the CP.

•••𝑃(𝑀) : probability distr.
𝑉(𝑀) : effective potential
𝑀 : order parameter

1st tr.

Ising Model• Sign of 𝑁 3 𝑐 is flipped at the CP.

Ising Model•••𝑃(𝑀) : probability distr.
𝑉(𝑀) : effective potential
𝑀 : order parameter• 𝑁 4 𝑐 changes discontinuously at the CP.

Reviews:
Asakawa, MK, PPNP (’16)
Bluhm, MK+, NPA 1003 (’20)
MK, Esumi, Nonaka, JPS journal, 2021/8

temperature

LHC

future facilities
ex. J-PARC @ Tokaibaryon chemical potential

Review: Asakawa, MK, PPNP 90 (2016)
STAR, PRL105 (2010)

Detector

Cumulants

•••𝑃(𝑀) : probability distr.
𝑉(𝑀) : effective potential
𝑀 : order parameter

1st tr.

Ising Model• Sign of 𝑁 3 𝑐 is flipped at the CP.

Asakawa, Ejiri, MK, PRL ’09

Geometric interpretation on the signs
Fluctuations 𝑁𝐵2 𝑐diverge at the QCD-CP.

Themodynamic Relation

Sign of 𝑁𝐵3 𝑐 can distinguishnear and away sides!

Asakawa, Ejiri, MK, PRL ’09

Once negative 𝑁𝐵3 𝑐 is established, it is evidences that
(1) cB has a peak structure in the QCD phase diagram.
(2) Hot matter beyond the peak is created in the collisions.

•No dependence on any specific models.
•Just the sign! No normalization (such as by Nch).

suppression?

STAR, 2001.06419

 Nonzero and non-Poissonian cumulants
are experimentally established.

initialfluctuations

Critical slowing
down around CP
diffusion offluctuations

Proper understanding of the time evolution offluctuations is indispensable.

 Higher order cumulants
in diffusion master equation

MK+ (2014); MK (2015)

 2nd order cumulant near CP
in stochastic diffusion equation
sakaida, Asakawa, Fujii, MK, 2018

 Non-monotonic Δ𝑦 dependence can emerge
reflecting the dynamical evolution.

Detector-response correctionMK, Asakawa, 2012
MK, 2016
MK, Luo, 2017
Nonaka, MK, Esumi, 2017
Nonaka, MK, Esumi, 2018

Pileup correction

true

Nonaka, MK, Esumi, 2020

Closer look at Exp. data
Nonaka, MK, Esumi, in prep.

pileupKiyohara, MK, Ejiri, Kanaya, arXiv:2108.00118
See also
WHOT-QCD Collab.
PTEP 2021 (2021) 013B08; Phys.Rev.D 101 (2020) 05450

 Columbia plot
= order of phase tr. at 𝜇 = 0

 Example
Phase diagram inheavy-quark region

CriticalPoint

Various orders of phase transition with variation of 𝑚𝑞 .

Ising Model•••𝑃(𝑀) : probability distr.
𝑉(𝑀) : effective potential
𝑀 : order parameter• 𝑁 4 𝑐 changes discontinuously at the CP.

Binder Cumulant

Ising Model𝑉=∞ Sudden change of 𝐵4 at the CP is smeared by finite 𝑉 effect.
 𝐵4 obtained for various 𝑉 has crossing at 𝑡 = 0.
 At the crossing point, 𝐵4 = 1.604 in 𝑍2 universality class.

Binder Cumulant

Ising Model𝑉=∞
smaller 𝑉 Sudden change of 𝐵4 at the CP is smeared by finite 𝑉 effect.
 𝐵4 obtained for various 𝑉 has crossing at 𝑡 = 0.
 At the crossing point, 𝐵4 = 1.604 in 𝑍2 universality class.

 Light-quark region
Kuramashi, Nakamura, Ohno, Takeda, ’20

 Heavy-quark region
Cuteri , Philipsen , Schön, Sciarra, ’21

𝑁𝑡 = 12, 𝐿𝑇 < 2.7

 Statistically-significant deviation of thecrossing point from the 3d-Ising value.
Too large finite-V effects?

𝑁𝑡 = 10, 𝐿𝑇 ≤ 6

 Coarse lattice: 𝑁𝑡 = 4
 But large spatial volume:
𝐿𝑇 = 𝑁𝑠 / 𝑁𝑡 ≤ 12

 Hopping-param. (~1/𝑚𝑞 ) expansion
 Monte-Calro with LO action
 High statistical analysis

Simulation params.

Z2𝐿𝑇 ≥ 9𝐿𝑇 ≥ 8
 𝐵4 and 𝜈 are consistent with Z2 universality classonly when 𝐿𝑇 ≥ 9 data are used for the analysis.

 Effective potential at the CP Scaling of order parameterZ2 scaling is well established Critical points appear many places in QCD at nonzero
temperature.
 Two experimental tools for the search for the CPs:
 Relativistic heavy-ion collisions
 Lattice QCD numerical simulations
 Various studies are ongoing using both real and virtual
experiments.