Nf=3 QCD - - PowerPoint PPT Presentation

Nf=3 QCD 相図

中村 宜文

理化学研究所 計算科学研究機構

２０１５年９月５日 「有限温度密度系の物理と格子ＱＣＤシミュレーション」研究会

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共同研究者

• 金 暁勇（アルゴンヌ国立研究所）
• 藏増 嘉伸（筑波大学/理化学研究所）
• 武田 真滋（金沢大学/理化学研究所）
• 宇川 彰（理化学研究所）

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内容

• QCD の相図の概観
• phase diagram for Nf = 3, µ = 0
• Critical endpoint of finite temperature phase transition for

three flavor QCD, Phys. Rev. D 91, 014508 (2015)

• phase diagram for Nf = 3, µ 0
• Curvature of the critical line on the plane of quark

chemical potential and pseudo scalar meson mass for three-flavor QCD, arXiv:1504.00113[hep-lat]

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QCD の相図の概観

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有限温度有限密度（物理点）

密度 温度

初期宇宙 クォークグルーオンプラズマ相

カラー超伝導相

ハドロン相

陽子・中性子・・

臨界終点

重イオン衝突実験

中性子星

5 / 40

3D φ4 theory

g g T T O(4) O(4)

U(2)xU(2)/U(2)

O(4) 1st order O(4)

strong U(1)A anomaly weak U(1)A anomaly Basile et al. (hep-lat/0509018)

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Columbia plot（possible）

standard scenario alternative scenario (weak U(1)A anomaly) Kanaya (arXiv:1012.4247[hep-lat])

7 / 40

phase diagram for Nf = 3, µ = 0

8 / 40

Motivation

• Critical endpoint (CEP) obtained with staggered and Wilson type

fermions is inconsistent. → Results in the continuum limit is necessary

mπ at the endpoint at µ = 0 (bottom-left corner of Columbia plot) Nt action mE

π [MeV]

4 unimproved staggered 260 de Forcrand, 6 unimproved staggered 150 Philipsen ’07 4 p4-improved staggered 70 Karsch et al. ’03 6 stout-improved staggered ≲ 50 Endr˝

• di et al. ’07

6 HISQ ≲ 50 Ding et al. ’11,...’15 4 unimproved Wilson ∼ 1100 Iwasaki et al. ’96

• Nf = 3 study is a stepping stone
• curvature of critical surface
• to the physical point

We determine CEP on ml = ms line with clover fermions in the continuum limit

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quark mass chemical potential temperature physical world 1st order transition surface crossover surface 2nd order line s i g n p r

• b

l e m s e r i

• u

s

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quark mass chemical potential temperature physical world 1st order transition surface crossover surface 2nd order line s i g n p r

• b

l e m s e r i

• u

s

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quark mass chemical potential temperature physical world 1st order transition surface crossover surface 2nd order line s i g n p r

• b

l e m s e r i

• u

s

10 / 40

Distinguishing between 1st, 2nd and crossover

criterion first order second order crossover distribution double peak single peak singe peak χpeak ∝ Nd

l

∝ Nγ/ν

l

• β(χpeak) − βc

∝ N−d

l

∝ N−1/ν

l

• kurtosis at Nl → ∞

K= -2

• 2 < K < 0
• scaling might work with wrong exponents near CEP
• peaks in histgram might emerge only at large Nl on weak 1st order
• K does not depend on volume at 2nd order phase transition point

M = N−β/ν

l

fM(tN1/ν

l

) K + 3 = B4(M) = N−4β/ν

l

fM4(tN1/ν

l

) [ N−2β/ν

l

fM2(tN1/ν

l

) ]2 = fB(tN1/ν

l

)

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Method to determine CEP

• determine the transition point (peak position of susceptibility)
• determine kurtosis at transition point at each spatial lattice size
• find intersection point of kurtosis by fit, KE + aN1/ν

l

(β − βE)

E 1st order crossover Kt

x x K K

• 2

light heavy

V>V>V

• interpolate/extrapolate √

t0mPS,t measured at transition point to βE

• extrapolate √

t0mPS,E to the continuum limit

• use scale determined from Wilson flow 1/ √

t0 = 1.347(30) GeV [Borsanyi et al. ’12]

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Higher moments

i-th derivative of ln Z with respect to control parameter c: E = ∂ ln Z

∂c

• Variance

V = ∂2 ln Z

∂c2 = σ2

• Skewness ( e.g. right-skewed → S > 0 , left-skewed

→ S < 0)

S = 1

σ3 ∂3 ln Z ∂c3

• Kurtosis( e.g. Gaussian → K = 0, 2δ func. → K = −2 )

K = 1

σ4 ∂4 ln Z ∂c4 = B4 − 3

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Simulations

• action: Iwasaki gluon + Nf = 3 clover (non perturbative cSW,

degenerate)

• observables
• gauge action density, G
• plaquette, P
• Polyakov loop, L
• chiral condensate, Σ
• and their higher moments
• temporal lattice size Nt = 4, 6, 8
• statistics: O(100K) traj
• preliminary Nt = 10
• statistics: O(1K) traj

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plaquette at β = 1.60, Nt = 4

0.5 1 1.5 2 2.5 3 3.5 4 susceptibility of 𝑄 𝑂𝑡 = 6 𝑂𝑡 = 8 𝑂𝑡 = 10 𝑂𝑡 = 12 𝑂𝑡 = 16 0.1432 0.1433 0.1434 0.1435 0.1436 0.1437

• 1.8
• 1.6
• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 kurtosis of 𝑄 𝜆

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plaquette at β = 1.65, Nt = 4

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 susceptibility of 𝑄 𝑂𝑡 = 6 𝑂𝑡 = 8 𝑂𝑡 = 10 𝑂𝑡 = 12 𝑂𝑡 = 16 0.141 0.1411 0.1412 0.1413 0.1414 0.1415

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 kurtosis of 𝑄 𝜆

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Kurtosis intersection at Nt = 4

• 2
• 1.5
• 1
• 0.5

𝐿𝑢 𝑂𝑚 = 10, 𝐻 𝑂𝑚 = 12, 𝐻 𝑂𝑚 = 16, 𝐻 𝑂𝑚 = 10, 𝑄 𝑂𝑚 = 12, 𝑄 𝑂𝑚 = 16, 𝑄 𝑂𝑚 = 10, 𝑀 𝑂𝑚 = 12, 𝑀 𝑂𝑚 = 16, 𝑀 0.55 0.60 0.65 0.70 0.75 0.80 0.85 󰠌𝑢0 𝑛PS,t 0.15 0.16 0.17 0.18 0.19 1.600 1.610 1.620 1.630 1.640 1.650 1.660 1.670 󰠌𝑢0 𝑈t 𝛾 17 / 40

Kurtosis intersection at Nt = 4

• 2
• 1.5
• 1
• 0.5

𝐿𝑢 𝑂𝑚 = 10, Σ 𝑂𝑚 = 12, Σ 𝑂𝑚 = 16, Σ 0.55 0.60 0.65 0.70 0.75 0.80 0.85 󰠌𝑢0 𝑛PS,t 0.15 0.16 0.17 0.18 0.19 1.600 1.610 1.620 1.630 1.640 1.650 1.660 1.670 󰠌𝑢0 𝑈t 𝛾 18 / 40

γ/ν v.s. β

0.5 1 1.5 2 2.5 3 1.6 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76 1.77 𝑐(= 𝛿⁄𝜈) 𝛾 𝛾𝐹 at 𝑂𝑢 = 4 𝛾𝐹 at 𝑂𝑢 = 6 𝛾𝐹 at 𝑂𝑢 = 8 𝑂𝑢 = 4, 𝐻 𝑂𝑢 = 4, 𝑀 𝑂𝑢 = 4, Σ 𝑂𝑢 = 6, 𝐻 𝑂𝑢 = 6, 𝑀 𝑂𝑢 = 6, Σ 𝑂𝑢 = 8, 𝐻 𝑂𝑢 = 8, 𝑀 𝑂𝑢 = 8, Σ 3D 𝑎2 3D 𝑃(2) 3D 𝑃(4)

χmax = aNγ/ν

l

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continuum extrapolation for √ t0mPS,E

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.01 0.02 0.03 0.04 0.05 0.06 0.07 󰠌𝑢0 𝑛PS, E 1⁄𝑂 2

𝑢

crossover 1st order SU(3) sysmmetric point Σ G, P, L

▲ : √ t0mphy;sym

PS

= √ t0 √ (m2

π + 2m2 K)/3 ∼ 0.305

mPS,E = 304(7)(14)(7) MeV

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continuum extrapolation for √ t0TE

0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.01 0.02 0.03 0.04 0.05 0.06 0.07 󰠌𝑢0 𝑈E 1⁄𝑂 2

𝑢

Σ G, P, L

TE = 131(2)(1)(3) MeV

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Summary at Nt = 4, 6, 8

• kurtosis intersection analysis is consistent with χmax

analysis

• results at Nt = 4 is out of scaling region
• √

t0mPS,E in the continuum limit is smaller than the SU(3) sysmmetric point, mPS,E/mphys,sym

PS

= 0.739(17)(34)(17)

• further studies at larger temporal sizes to obtain

conclusive results are needed

• Phys. Rev. D 91, 014508 (2015)

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Σ at Nt = 10 (1/3, preliminary)

2000 4000 6000 8000 10000 12000 14000 0.1386 0.1388 0.139 0.1392 0.1394 0.1396 0.1398 0.14 0.1402 pbpz sus κ β=1.76 Ns=16 β=1.77 Ns=16 β=1.77 Ns=20 β=1.78 Ns=16 β=1.78 Ns=20 β=1.78 Ns=24 β=1.79 Ns=16 β=1.79 Ns=20 β=1.79 Ns=24 β=1.80 Ns=16 β=1.80 Ns=20

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Σ at Nt = 10 (2/3, preliminary)

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 0.1386 0.1388 0.139 0.1392 0.1394 0.1396 0.1398 0.14 0.1402 pbpz krt κ Z2 β=1.76 Ns=16 β=1.77 Ns=16 β=1.77 Ns=20 β=1.78 Ns=16 β=1.78 Ns=20 β=1.78 Ns=24 β=1.79 Ns=16 β=1.79 Ns=20 β=1.79 Ns=24 β=1.80 Ns=16 β=1.80 Ns=20

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Σ at Nt = 10 (3/3, preliminary)

0.25 0.30 0.35 󰠌𝑢0 𝑛PS,t 0.09 0.10 0.11 0.12 0.13 1.760 1.770 1.780 1.790 1.800 󰠌𝑢0 𝑈t 𝛾

assuming βE = 1.78(1)

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continuum extrapolation and results at Nt = 10 (preliminary)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.01 0.02 0.03 0.04 0.05 0.06 0.07 󰠌𝑢0 𝑛PS, E 1⁄𝑂 2

𝑢

crossover 1st order SU(3) sysmmetric point Σ G, P, L 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.01 0.02 0.03 0.04 0.05 0.06 0.07 󰠌𝑢0 𝑈E 1⁄𝑂 2

𝑢

Σ G, P, L

• assuming βE = 1.78(1) at Nt = 10
• excluding results at Nt = 10 from continuum

extrapolation

• TE would not change very much
• mPS,E may become smaller than results at smaller Nt

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Summary (µ = 0)

We have investigated the critical endpoint of QCD with clover fermions and determined the critical endpoint by using the intersection points of the Binder cumulants and extrapolated to the continuum limit

• TE in the continuum limit would not change very much

TE ≈ 130 MeV

• mPS,E in the continuum limit may become smaller than

results at smaller Nt mPS,E < 304(7)(14)(7) MeV? mPS,E/mphys,sym

PS

< 0.739(17)(34)(17)?

• we are doing further studies with high statistics at larger

temporal sizes to obtain conclusive results

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phase diagram for Nf = 3, µ 0

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Motivation

* QCD critical point crossover 1rst ∞ Real world X Heavy quarks mu,d ms µ QCD critical point DISAPPEARED crossover 1rst ∞ Real world X Heavy quarks mu,d ms µ

de Forcrand and Philipsen (JHEP 0701:077,2007)

We investigate curvature

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Finite density simulation

Z =

∫

DUe−Sg(det D(µ))Nf =

∫

DUe−Sg| det D(µ)|NfeiNfθ

= ∫

DUe−Sg det D†(µ)Nf/2 det D(µ)Nf/2eiNfθ

= ∫

DUe−Sg[det D†(µ)D(µ)]Nf/2eiNfθ

• generate gauge field configurations with weight:

e−Sg+ln det[D†(µ)D(µ)]Nf /2

• aµ = 0.1
• at Nt = 6, Nl = 8, 10, 12, β = 1.70 − 1.77
• statistics: O(10,000) - O(100,000) traj.

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Finite density simulation

• reweight with phase θ

⟨O⟩ = ⟨OeiNfθ⟩ ⟨cos(Nfθ)⟩

• cost reduction to calculate θ : (but cost is still expensive, O(n3))

det [ A B C D ] = det[A] det[D] det[1 − D−1CA−1B]

10 100 1000 10000 100000 1000 10000 100000 computation time [s] lattice size naive, 36 node naive,144 node reduction, 16 node reduction, 36 node reduction, 64 node reduction,144 node 31 / 40

cos(3θ)

0.2 0.4 0.6 0.8 1 0.138 0.139 0.14 0.141 0.142 <cos(Nfθ)>|| κ β=1.70, Ns=8 Ns=10 Ns=12 β=1.73, Ns=8 Ns=10 Ns=12 β=1.75, Ns=8 Ns=10 Ns=12 β=1.77, Ns=8 Ns=10 Ns=12

32 / 40

reweighting

We calculate det DW(m′

0, µ′)

det DW(m0, µ)

= exp [

ln det DW(m′

0, µ′)

det DW(m0, µ)

] = exp [

∞

∑

j,k=0

∆j

m0∆k µ

j!k!

( ∂ ∂m0 )j( ∂ ∂µ )k

ln det DW(m0, µ)

− ln det DW(m0, µ) ]

with

∆m0 = m′

0 − m0

∆µ = µ′ − µ

for the reweighting, where we use m0 = 1/(2κ) − 4 to see easily mass derivative of D.

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χ and K

50 100 150 200 250 susceptibility aµ=0 aµ=0.1 aµ=0.2 Ns=8 Ns=10 Ns=12 0.1412 0.1413 0.1414 0.1415 0.1416

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 kurtosis κ β=1.70 aµ=0 aµ=0.1 aµ=0.2 10 20 30 40 50 60 70 susceptibility aµ=0 aµ=0.1 aµ=0.19 Ns=8 Ns=10 Ns=12 0.1400 0.1401 0.1402 0.1403 0.1404 0.1405

• 1.6
• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

kurtosis κ β=1.73 aµ=0 aµ=0.1 aµ=0.19 5 10 15 20 25 30 susceptibility aµ=0 aµ=0.1 aµ=0.18 Ns=8 Ns=10 Ns=12 0.1392 0.1393 0.1394 0.1395 0.1396 0.1397 0.1398 0.1399

• 1.5
• 1
• 0.5

0.5 1 kurtosis κ β=1.75 aµ=0 aµ=0.1 aµ=0.18 4 6 8 10 12 14 susceptibility aµ=0 aµ=0.1 aµ=0.16 Ns=8 Ns=10 Ns=12 0.1384 0.1385 0.1386 0.1387 0.1388 0.1389 0.1390 0.1391

• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 kurtosis κ β=1.77 aµ=0 aµ=0.1 aµ=0.16

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K intersection at aµ = 0.00

• 2
• 1.5
• 1
• 0.5

1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 kurtosis β aµ=0.00 Ns=8 Ns=10 Ns=12 CEP 3-dim Z2

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K intersection at aµ = 0.10

• 2
• 1.5
• 1
• 0.5

1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 kurtosis β aµ=0.10 Ns=8 Ns=10 Ns=12 CEP 3-dim Z2

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K intersection at aµ = 0.15

• 2
• 1.5
• 1
• 0.5

1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 kurtosis β aµ=0.15 Ns=8 Ns=10 Ns=12 CEP 3-dim Z2

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K intersection at aµ = 0.19

• 2
• 1.5
• 1
• 0.5

1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 kurtosis β aµ=0.19 Ns=8 Ns=10 Ns=12 CEP 3-dim Z2

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curvature

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.05 0.1 0.15 0.2 0.25 0.3 0.35 (mPS(µ)/mPS(0))2 µ/(πTE(0)) mPSt0

1/2=0.55

mPSt0

1/2=0.65

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Summary

• We have investigated the critical endpoint of QCD with clover

fermions

• We have determined the critical endpoint by using the

intersection points of the Binder cumulants

• continuum extrapolation aµ = 0

mPS,E < 304(7)(14)(7) MeV? mPS,E/mphys,sym

PS

< 0.739(17)(34)(17)?

• for aµ 0, we have found ∂mE/∂µ > 0 at heavier quark

mass

40 / 40

Backup slides

1 / 6

Columbia plot

• inconsistent results: Wilson and staggered type fermion

ms mud

∞ ∞ N = 2

F

N = 3

F

N = 1

F pure gauge

1st

• rder

1st

• rder

2nd order crossover

≈140 ≈150 * ≈400 MeV ≈250MeV ≈100 ≈1.2GeV ≈50

• •
• ms

*

• Wilson

staggered

2 / 6

Finite temperature phase transition

0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.141 0.1415 0.142 0.1425 0.143 0.1435 0.144 0.1445 plaquette 𝜆 63 × 4 63 × 12

• Plaquette v.s. κ at lowest β (= 1.60)
• no bulk phase transition

3 / 6

1st order phase transition and crossover (like)

β = 1.60 and κ = 0.14345 on 103 × 4, clear two states,

K ∼ −1.5

• β = 1.70 and κ = 0.13860 on 103 × 4, one state, K ∼ −0.5
• 4 / 6

5 10 15 20 25 30 35 susceptibility of 𝐻 𝑂𝑡 = 6 𝑂𝑡 = 8 𝑂𝑡 = 10 𝑂𝑡 = 12 𝑂𝑡 = 16 0.1432 0.1433 0.1434 0.1435 0.1436 0.1437

• 1.8
• 1.6
• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 kurtosis of 𝐻 𝜆 0.5 1 1.5 2 2.5 3 3.5 4 4.5 susceptibility of 𝑀 𝑂𝑡 = 6 𝑂𝑡 = 8 𝑂𝑡 = 10 𝑂𝑡 = 12 𝑂𝑡 = 16 0.1432 0.1433 0.1434 0.1435 0.1436 0.1437

• 1.8
• 1.6
• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 kurtosis of 𝑀 𝜆 50 100 150 200 250 300 350 400 450 500 susceptibility of Σ 𝑂𝑡 = 6 𝑂𝑡 = 8 𝑂𝑡 = 10 𝑂𝑡 = 12 𝑂𝑡 = 16 0.1432 0.1433 0.1434 0.1435 0.1436 0.1437

• 1.8
• 1.6
• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 kurtosis of Σ 𝜆 1 1.5 2 2.5 3 3.5 4 4.5 susceptibility of 𝐻 𝑂𝑡 = 6 𝑂𝑡 = 8 𝑂𝑡 = 10 𝑂𝑡 = 12 𝑂𝑡 = 16 0.141 0.1411 0.1412 0.1413 0.1414 0.1415

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 kurtosis of 𝐻 𝜆 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 susceptibility of 𝑀 𝑂𝑡 = 6 𝑂𝑡 = 8 𝑂𝑡 = 10 𝑂𝑡 = 12 𝑂𝑡 = 16 0.141 0.1411 0.1412 0.1413 0.1414 0.1415

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 kurtosis of 𝑀 𝜆 10 15 20 25 30 35 40 45 50 susceptibility of Σ 𝑂𝑡 = 6 𝑂𝑡 = 8 𝑂𝑡 = 10 𝑂𝑡 = 12 𝑂𝑡 = 16 0.141 0.1411 0.1412 0.1413 0.1414 0.1415

• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 kurtosis of Σ 𝜆

5 / 6

Critical endpoint at Nt = 6, 8

• 2
• 1.5
• 1
• 0.5

𝐿𝑢 𝑂𝑚 = 10, 𝐻 𝑂𝑚 = 12, 𝐻 𝑂𝑚 = 16, 𝐻 𝑂𝑚 = 10, 𝑄 𝑂𝑚 = 12, 𝑄 𝑂𝑚 = 16, 𝑄 𝑂𝑚 = 10, 𝑀 𝑂𝑚 = 12, 𝑀 𝑂𝑚 = 16, 𝑀 0.45 0.50 0.55 0.60 0.65 󰠌𝑢0 𝑛PS,t 0.12 0.13 0.14 0.15 0.16 1.700 1.710 1.720 1.730 1.740 1.750 1.760 1.770 󰠌𝑢0 𝑈t 𝛾

• 2
• 1.5
• 1
• 0.5

𝐿𝑢 𝑂𝑚 = 16, 𝐻 𝑂𝑚 = 20, 𝐻 𝑂𝑚 = 24, 𝐻 𝑂𝑚 = 16, 𝑄 𝑂𝑚 = 20, 𝑄 𝑂𝑚 = 24, 𝑄 𝑂𝑚 = 16, 𝑀 𝑂𝑚 = 20, 𝑀 𝑂𝑚 = 24, 𝑀 0.35 0.40 0.45 󰠌𝑢0 𝑛PS,t 0.11 0.12 0.13 1.700 1.710 1.720 1.730 1.740 1.750 1.760 1.770 󰠌𝑢0 𝑈t 𝛾

• 2
• 1.5
• 1
• 0.5

𝐿𝑢 𝑂𝑚 = 10, Σ 𝑂𝑚 = 12, Σ 𝑂𝑚 = 16, Σ 0.45 0.50 0.55 0.60 0.65 󰠌𝑢0 𝑛PS,t 0.12 0.13 0.14 0.15 0.16 1.700 1.710 1.720 1.730 1.740 1.750 1.760 1.770 󰠌𝑢0 𝑈t 𝛾

• 2
• 1.5
• 1
• 0.5

𝐿𝑢 𝑂𝑚 = 16, Σ 𝑂𝑚 = 20, Σ 𝑂𝑚 = 24, Σ 0.35 0.40 0.45 󰠌𝑢0 𝑛PS,t 0.11 0.12 0.13 1.700 1.710 1.720 1.730 1.740 1.750 1.760 1.770 󰠌𝑢0 𝑈t 𝛾

Nt = 6 Nt = 8

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