exploring the qcd phase diagram with imaginary chemical ......anirban lahiri 1 jishnu goswami...
TRANSCRIPT
Exploring the QCD Phase diagram with imaginary chemical
potential with HISQ action
Collaborators: Frithjof Karsch, Christian Schmidt, and Anirban Lahiri
�1
Jishnu Goswami
Universität Bielefeld
Plan of the talk
• Introduction
• Status on chiral phase transition with HISQ at 𝜇 =0
• Status on chiral phase transition in the RW plane.
• Results and discussions.
!2
Introduction
!3
2nd 3d Ising
mu,d ms
11st tr.
1st tr.
1Nf
= 1
Nf = 2
0
�✓⇡3
◆2
✓µT
◆2
tric
tric
1stZ(2) Z(2)
1st
phys. point
Nf = 3 crossover
O. Philipsen and C. Pinke. Phys. Rev. D93, 114507, 2016.
Central Question:Nature of the chiral symmetry restoring transition at 𝜇=0 at the chiral limit??
Possible scenario of extended 3d Columbia plot
The 1st order region is expected to be largest in the RW plane(𝜇/T=(2k+1)𝜋/3). Thus critical mass in the RW plane puts a bound on the critical mass at 𝜇=0.
Does a 1st order chiral symmetry restoring transition exist at 𝜇=0 below a certain critical quark mass (mcri) ??
Chiral transition for zero chemical potential with HISQ
• HotQCD results on chiral phase transition, [ ]
!4
Nf =3 : 1st order phase transition ruled out for 230 MeV > m𝜋 >80 MeV. Bound on critical pion mass is given as, m𝜋
cr ≲50 MeV from the scaling analysis.
μ = 0
mud
ms
PureGauge
1st
1st
crossover
Nf=
3phys.point
Z(2)
Z(2)
Nf = 2
mtrics
Nf
=1
O(4)?U(2)L ¢ U(2)R/U(2)V ?
chiral limit Nf = 2
Nf =2+1 : No hint of 1st order phase transition for m𝜋 >55 MeV. chiral transition is most likely 2nd order O(N) rather than Z(2).
A. Lahiri et. al. , QM 2018, arXiv:1807.05727
Bazavov et. al. PRD 95, 074505 (2017)
m𝜋﹥1 GeV, for the ‘heavy quark mass RW transition’
Standard staggered action: m𝜋~400 MeV (N𝝉=4) Standard Wilson action: m𝜋~930 MeV(N𝝉=4) m𝜋~680 MeV (N𝝉=6) 1st order end point (of the line of 1st order RW transitions) exist already for 𝜇/T=𝜋/3 and mcri >mphy.
The results are strongly fermion discretization scheme and cut-off(N𝝉) dependent.
!5
P. de Forcrand et. al, PRL 105, 152001(2010), Owe Philipsen et. al, PRD 89, 094504(2014), Christopher Czaban et al, PRD 93, 054507 (2016)
‘small quark mass RW transition’
Studies in the RW plane Mostly with unimproved actions
(Nf = 2)
Possible scenario of RW end point—plenary by S. Mukherjee
• Stout improved staggered fermions(Nf =2+1): At the physical quark mass point(m𝜋~135 MeV) a 2nd order transition in the 3d-Ising universality class happens instead of a 1st order at the RW endpoint.
No 1st order end point (of the line of 1st order RW transitions) for m𝜋 >50 MeV.
• HISQ(Nf =2): Order of the phase transition at physical point is not clear(large cut-off effects) .
!6
C. Bonati et. al,PRD 93, 074504 (2016)
L.K.Wu, et al. PRD 97,114514(2018)
C. Bonati et. al,arXiv:1807.02106 [hep-lat]
Studies in the RW plane Very recent studies with improved actions,
Studies with HISQ in the RW plane
!7
Mq = DHISQ(μf ) + mq
Action,
Simulation details,
(MeV)
8 4 1/27 135
12 4 1/27 135
16 4 1/27,1/40,1/60
135, 110, 90
24 4 1/27,1/40 135, 110
Nσ Nτml
ms
mπ We vary 𝛽 in the range [5.850-6.038],
Nf = 2 + 1,μT
=π3
We work on the 2nd RW plane
Z(T, μ) = ∫ [𝒟U]det[Mud(μf )]1/2det[Ms(μf )]1/4exp[−SG]
Generally we generated 20k trajectory per 𝛽 value away
from 𝛽c and 80k trajectory near 𝛽c
∼ Tc ± 0.1Tccorresponds to,
Ising endpoint of a first order line
!8
Heff(t, ξ) = tℰ + hℳEffective Ising Hamiltonian which
defines the universal critical behaviour of the system
temperature likefield
energy likeoperator
magnetic field like
magnetization like operator
(order parameter)
Corresponding critical behaviour of QCD in 2nd RW plane [Z(2) transformation],
under Z(2) transformation,ℰ → ℰℳ → − ℳ
Im L → − Im LRe L → Re L
order parameter
energy like
i.e. at μ = μRW
limh!0
limV!1
hIm Li ⌘ limV!1
h|Im L|i =(0, if � < �c
non-zero, if � > �c<latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">AAACv3icdVFNj9MwEHXC1xK+Chy5WFQgDkvllGq7SAta4AISh0Wi3ZXqKnLcSWrVcYLtrAjZ8CM5IPFvcJogPlRGsvzmzXua8TgupDCWkB+ef+nylavX9q4HN27eun1ncPfe3OSl5jDjucz1WcwMSKFgZoWVcFZoYFks4TTevGnrp+egjcjVR1sVsMxYqkQiOLOOigbfqRRZVK+pzTFpcJfN24wKldiqoZKpVAJ+l319j6nuEgqfSnG+Q417+UUrv+jlL3BAY0iFqrmb1DQBdkH2H2Nq4bOtRYKdLwbL8NH2ijilW01XV7l6+gV03uzjHZaXvy0U1KpvEQ2GZPT88GA8OcBkRMg0HIctGE8nzyY4dEwbQ9THSTT4Rlc5LzNQlktmzCIkhV3WTFvBJTQBLQ0UjG9YCgsHFcvALOvt/hv8yDErnOTaHWXxlv3TUbPMmCqLnTJjdm3+rbXkrtqitMnhshaqKC0o3jVKSondwtvPxCuhgVtZOcC4Fm5WzNdMM27dlwduCb9eiv8P5uNRSEbhh8nw+HW/jj30AD1ET1CIpugYvUUnaIa4d+TF3saT/is/9ZVfdFLf6z330V/hVz8BCrPZDg==</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">AAACv3icdVFNj9MwEHXC1xK+Chy5WFQgDkvllGq7SAta4AISh0Wi3ZXqKnLcSWrVcYLtrAjZ8CM5IPFvcJogPlRGsvzmzXua8TgupDCWkB+ef+nylavX9q4HN27eun1ncPfe3OSl5jDjucz1WcwMSKFgZoWVcFZoYFks4TTevGnrp+egjcjVR1sVsMxYqkQiOLOOigbfqRRZVK+pzTFpcJfN24wKldiqoZKpVAJ+l319j6nuEgqfSnG+Q417+UUrv+jlL3BAY0iFqrmb1DQBdkH2H2Nq4bOtRYKdLwbL8NH2ijilW01XV7l6+gV03uzjHZaXvy0U1KpvEQ2GZPT88GA8OcBkRMg0HIctGE8nzyY4dEwbQ9THSTT4Rlc5LzNQlktmzCIkhV3WTFvBJTQBLQ0UjG9YCgsHFcvALOvt/hv8yDErnOTaHWXxlv3TUbPMmCqLnTJjdm3+rbXkrtqitMnhshaqKC0o3jVKSondwtvPxCuhgVtZOcC4Fm5WzNdMM27dlwduCb9eiv8P5uNRSEbhh8nw+HW/jj30AD1ET1CIpugYvUUnaIa4d+TF3saT/is/9ZVfdFLf6z330V/hVz8BCrPZDg==</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit>
!9
Corresponding critical behaviour of QCD at 2nd RW plane [Z(2) transformation],
Im L → − Im LRe L → Re L
order parameter
energy like
i.e. at μ = μRW
limh!0
limV!1
hIm Li ⌘ limV!1
h|Im L|i =(0, if � < �c
non-zero, if � > �c<latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit>
!9
Corresponding critical behaviour of QCD at 2nd RW plane [Z(2) transformation],
Im L → − Im LRe L → Re L
order parameter
energy like
i.e. at μ = μRW
limh!0
limV!1
hIm Li ⌘ limV!1
h|Im L|i =(0, if � < �c
non-zero, if � > �c<latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">AAACv3icdVFNj9MwEHXC1xK+Chy5WFQgDkvllGq7SAta4AISh0Wi3ZXqKnLcSWrVcYLtrAjZ8CM5IPFvcJogPlRGsvzmzXua8TgupDCWkB+ef+nylavX9q4HN27eun1ncPfe3OSl5jDjucz1WcwMSKFgZoWVcFZoYFks4TTevGnrp+egjcjVR1sVsMxYqkQiOLOOigbfqRRZVK+pzTFpcJfN24wKldiqoZKpVAJ+l319j6nuEgqfSnG+Q417+UUrv+jlL3BAY0iFqrmb1DQBdkH2H2Nq4bOtRYKdLwbL8NH2ijilW01XV7l6+gV03uzjHZaXvy0U1KpvEQ2GZPT88GA8OcBkRMg0HIctGE8nzyY4dEwbQ9THSTT4Rlc5LzNQlktmzCIkhV3WTFvBJTQBLQ0UjG9YCgsHFcvALOvt/hv8yDErnOTaHWXxlv3TUbPMmCqLnTJjdm3+rbXkrtqitMnhshaqKC0o3jVKSondwtvPxCuhgVtZOcC4Fm5WzNdMM27dlwduCb9eiv8P5uNRSEbhh8nw+HW/jj30AD1ET1CIpugYvUUnaIa4d+TF3saT/is/9ZVfdFLf6z330V/hVz8BCrPZDg==</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit>
�0.02 �0.01 0.00 0.01 0.02�(ReL)
�0.04
�0.02
0.00
0.02
0.04
�(Im
L)
� = 5.925T<Tc
!9
Corresponding critical behaviour of QCD at 2nd RW plane [Z(2) transformation],
Im L → − Im LRe L → Re L
order parameter
energy like
i.e. at μ = μRW
limh!0
limV!1
hIm Li ⌘ limV!1
h|Im L|i =(0, if � < �c
non-zero, if � > �c<latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit><latexit sha1_base64="gRtvq7rDWgrqzr8hjBTQJw8TnZQ=">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</latexit>
�0.02 �0.01 0.00 0.01 0.02�(ReL)
�0.04
�0.02
0.00
0.02
0.04
�(Im
L)
� = 5.925T<Tc
�0.02 �0.01 0.00 0.01 0.02�(ReL)
�0.08�0.06�0.04�0.02
0.000.020.040.06
�(Im
L)
� = 5.990T>Tc
Finite size scaling and Z(2) universality class
!10
ut ∼ ctt, ut ∼ chhnear, T → Tc
t =T − Tc
Tc∼ β − βc
χh = z1Nγ/νσ fh(z0tN1/ν
σ )
χt = z2Nα/νσ ft(z0tN1/ν
σ )
f = fns + b−dfs(bytut, byhuh, b−1Nσ)Free energy,
Responsible for the universal critical behaviour
Susceptibility of |Im L| Susceptibility of |Re L|
< O > = ( . . . . )∂∂h
fs( . . . ) |h→o
χt = ( . . . . )∂2
∂t2fs( . . . ) |h→o
χh = ( . . . . )∂2
∂h2fs( . . . ) |h→o susceptibility of op
order parameter
specific heatuniversal functions
Finite size scaling and Z(2) universality class
!10
ut ∼ ctt, ut ∼ chhnear, T → Tc
t =T − Tc
Tc∼ β − βc
χh = z1Nγ/νσ fh(z0tN1/ν
σ )
χt = z2Nα/νσ ft(z0tN1/ν
σ )
f = fns + b−dfs(bytut, byhuh, b−1Nσ)Free energy,
Responsible for the universal critical behaviour
Susceptibility of |Im L| Susceptibility of |Re L|
< O > = ( . . . . )∂∂h
fs( . . . ) |h→o
χt = ( . . . . )∂2
∂t2fs( . . . ) |h→o
χh = ( . . . . )∂2
∂h2fs( . . . ) |h→o susceptibility of op
order parameter
specific heatuniversal functions
0
0.2
0.4
0.6
0.8
1
1.2
1.4
175 180 185 190 195 200 205 210
line: Z(2) scaling curveTc = 202.6(4) MeV
χ h
T [MeV]
Nσ=241612
8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
175 180 185 190 195 200 205 210
mπ ∼ 135 MeV
χ h
Finite size scaling and Z(2) universality class
!10
ut ∼ ctt, ut ∼ chhnear, T → Tc
t =T − Tc
Tc∼ β − βc
χh = z1Nγ/νσ fh(z0tN1/ν
σ )
χt = z2Nα/νσ ft(z0tN1/ν
σ )
f = fns + b−dfs(bytut, byhuh, b−1Nσ)Free energy,
Responsible for the universal critical behaviour
Susceptibility of |Im L| Susceptibility of |Re L|
< O > = ( . . . . )∂∂h
fs( . . . ) |h→o
χt = ( . . . . )∂2
∂t2fs( . . . ) |h→o
χh = ( . . . . )∂2
∂h2fs( . . . ) |h→o susceptibility of op
order parameter
specific heatuniversal functions
175 190 200 215T [MeV]
0.075
0.080
0.085
0.090
0.095
0.100
�R
eL
Ns = 12
Ns = 16
Ns = 24
mπ ∼ 135 MeV
χ t
0
0.2
0.4
0.6
0.8
1
1.2
1.4
175 180 185 190 195 200 205 210
line: Z(2) scaling curveTc = 202.6(4) MeV
χ h
T [MeV]
Nσ=241612
8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
175 180 185 190 195 200 205 210
mπ ∼ 135 MeV
χ h
Finite size scaling and Z(2) universality class
!11
ut ∼ ctt, ut ∼ chh near, T → Tc
t =T − Tc
Tc∼ β − βc
f = fns + b−dfs(bytut, byhuh, b−1Nσ)Free energy,
Responsible for the universal critical behaviour
are 3d Ising Exponentsα, γ, ν
“Susceptibility” and “specific heat” scale with
corresponding Z(2) finite size universal scaling functions
Finite size scaling and Z(2) universality class
!11
ut ∼ ctt, ut ∼ chh near, T → Tc
t =T − Tc
Tc∼ β − βc
f = fns + b−dfs(bytut, byhuh, b−1Nσ)Free energy,
Responsible for the universal critical behaviour
are 3d Ising Exponentsα, γ, ν
χh = z1Nγ/νσ fh(z0tN1/ν
σ )
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
-6 -4 -2 0 2 4
line: Z(2) scaling curveTc = 202.6(4) MeV
χ h N
σ−γ/ν
z=z0 Nσ1/ν (T-Tc)/Tc
Nσ=241612
8
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
-6 -4 -2 0 2 4
mπ ∼ 135 MeV
“Susceptibility” and “specific heat” scale with
corresponding Z(2) finite size universal scaling functions
Finite size scaling and Z(2) universality class
!11
ut ∼ ctt, ut ∼ chh near, T → Tc
t =T − Tc
Tc∼ β − βc
f = fns + b−dfs(bytut, byhuh, b−1Nσ)Free energy,
Responsible for the universal critical behaviour
are 3d Ising Exponentsα, γ, ν
χh = z1Nγ/νσ fh(z0tN1/ν
σ )
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
-6 -4 -2 0 2 4
line: Z(2) scaling curveTc = 202.6(4) MeV
χ h N
σ−γ/ν
z=z0 Nσ1/ν (T-Tc)/Tc
Nσ=241612
8
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
-6 -4 -2 0 2 4
mπ ∼ 135 MeV
χt = z2Nα/νσ ft(z0tN1/ν
σ )
�10 �5 0 5(T�Tc)
TcN 1/⌫
�
0.0450
0.0475
0.0500
0.0525
0.0550
0.0575
0.0600
0.0625
�tN
�↵/⌫
�
N�=12
N�=16
N�=24
mπ ∼ 135 MeV
“Susceptibility” and “specific heat” scale with
corresponding Z(2) finite size universal scaling functions
!12
Binder cumulants of order parameter |Im L|
175 190 200 215
2
3
4
BIm
L
Ns = 8
Ns = 12
190 200 215T (MeV)
binder cumulant of the imaginary part of LNs = 12
Ns = 16
190 200 215
Ns = 16
Ns = 24
Binder cumulants of Im L
Ising value=1.604
mπ ∼ 135 MeV
mπ ∼ 135 MeV
0
0.01
0.02
0.03
0.04
0.05
175 180 185 190 195 200 205 210 215
lines: Z(2) scaling fitTc = 200.8(3) MeV
<|Im
L|>
T [MeV]
Nσ=241612
0
0.01
0.02
0.03
0.04
0.05
175 180 185 190 195 200 205 210 215
mπ ∼ 135 MeV
0
0.05
0.1
0.15
0.2
0.25
-6 -4 -2 0 2 4 6
line: Z(2) scaling curveTc = 200.8(3) MeV
<|Im
L|>
Nσβ/ν
z=z0 Nσ1/ν (T-Tc)/Tc
Nσ=241612
8 0
0.05
0.1
0.15
0.2
0.25
-6 -4 -2 0 2 4 6
Universal Z(2) scaling of theorder parameter
Crossing of the Binder cumulants approaches
to the universal value in V → ∞
Quark mass dependence of RW transition
!13
No unusual change in the susceptibility of the order parameter with respect to pion mass upto, MeV mπ ∼ 90
Order of the transition seems to be unchanged ??
Susceptibility of |Im L|
Susceptibility of |Im L|
175 190 200 215T (MeV)
0.1
0.2
0.3
0.4
0.5
0.6
�|I
mL
|
ml/ms = 1/27
ml/ms = 1/40
ml/ms = 1/60
185 190 195 200 205 210T (MeV)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
�|I
mL
|
ml/ms = 1/27
ml/ms = 1/40
Nσ = 24
Nσ = 16
Chiral limit and RW transition
!14
⟨ψ̄ ψ⟩ ∼ const + m1/2
χdiscψ̄ψ ∼ m−1/2 For, T < Tc
Susceptibility of ⟨l̄l⟩
Sub. Chiral condensate
Δls = (ms/f 4k )(⟨l̄l⟩ − (ml /ms)⟨s̄s⟩)
175 190 200 220T [MeV]
10
20
30
40
50
�di
scl̄l
/T2
ml/ms = 1/27
ml/ms = 1/40
ml/ms = 1/60Goldstone effect (square root singularity) in below TRW is evident. chiral symmetry restoration at TRW ?
χdiscl̄l
175 180 185 190 195 200 205 210 215T[MeV]
5
10
15
20
25
30
�ls
ml/ms = 1/27
ml/ms = 1/40
ml/ms = 1/60
Nσ = 16
Nσ = 16
!15
Nf =2+1, m𝜋 ~135 MeV
Strong volume dependence of “subtracted chiral condensate” at fixed ml/ms below TRW
�0.06 �0.02 0.02 0.06 0.10�(h ̄ i)
�0.06�0.04�0.02
0.000.020.040.06
�(Im
L)
� = 5.975
175 180 185 190 195 200 205 210 215T(MeV)
5
10
15
20
25
30
�ls
N�=12
N�=16
N�=24
175 180 185 190 195 200 205 210 215T [MeV]
0.0
0.5
1.0
1.5
2.0
�d�
lsdT
N�=12
N�=16
N�=24
mixed chiral susceptibility sensitive to transition at the RW endpoint
T=201.85 MeVml /ms = 1/27mπ ∼ 135 MeV
mπ ∼ 135 MeV
Nσ = 16mπ ∼ 135 MeV
Conclusions• Our preliminary findings from calculations with the HISQ
action with physical pion mass suggest that the RW-end point is 2nd order and belongs to the Z(2) universality class. This is consistent with the earlier result found with the stout-improved staggered action.
• preliminary trends of m𝜋~110, 90 MeV results are also consistent with a 2nd order phase transition.
• RW transition and chiral phase transition may coincide in the chiral limit.
Calculations on larger lattices and smaller quark masses are ongoing.
!16
Conclusions• Our preliminary findings from calculations with the HISQ
action with physical pion mass suggest that the RW-end point is 2nd order and belongs to the Z(2) universality class. This is consistent with the earlier result found with the stout-improved staggered action.
• preliminary trends of m𝜋~110, 90 MeV results are also consistent with a 2nd order phase transition.
• RW transition and chiral phase transition may coincide in the chiral limit.
Calculations on larger lattices and smaller quark masses are ongoing.
!16
Back up slides
!17
!18
TRW
crossover
0⇡3µT
T Z(2) 2nd order(3d Ising)
Tpc
1st order
Intermediate quark mass
RW trans.
crossover
0⇡3µT
T
deconfinement 1st
orderTc
1st orderTriple point
1st orderRW trans.
Heavy quark mass
0⇡3µT
T 1st orderTriple point
Tpc
small quark mass
RW trans.
crossover
2nd orderend point
Chiral 1st order??
Different scenarios for different quark masses
TRW
TRW
TRW
Conjectured phase diagrams in the imaginary
chemical potential plane
1st order
Phases in the RW plane
• RW transition happens between two Z(3) sectors of the Polyakov loop. Hence, the order parameter can be the phase or the imaginary part of the Polyakov loop.
• In the RW plane, the 1st order region (for small mass) consists of three 1st order transitions, where high temperature RW transition meets two chiral phase transitions.
• The physical point which is crossover for 𝜇=0 can be 1st or 2nd order in the RW plane. So, our first goal is to confirm this issue and then going to the chiral limit to “search for a 1st order” transition.
!19
Nf = 3
Nf = 2<latexit sha1_base64="lDZ6GTcUZ43Dbrzla29lS8np/dM=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoMgHpbdNSTxIAQ86EkimAckS5idzCZjZh/MzAphyT948aCIV3/Fszf/xtlEQUULGoqqbrq7vJgzqSzr3cgtLC4tr+RXC2vrG5tbxe2dlowSQWiTRDwSHQ9LyllIm4opTjuxoDjwOG1747PMb99SIVkUXqtJTN0AD0PmM4KVllqXff/UKfSLJcs8qVWccgVZpmVVbcfOiFMtH5eRrZUMpfrRq40AoNEvvvUGEUkCGirCsZRd24qVm2KhGOF0WuglksaYjPGQdjUNcUClm86unaIDrQyQHwldoUIz9ftEigMpJ4GnOwOsRvK3l4l/ed1E+TU3ZWGcKBqS+SI/4UhFKHsdDZigRPGJJpgIpm9FZIQFJkoHlIXw9Sn6n7Qc07ZM+0qncQ5z5GEP9uEQbKhCHS6gAU0gcAN38ACPRmTcG0/G87w1Z3zO7MIPGC8fsdOQCA==</latexit><latexit sha1_base64="I/gpFOl+SGFJtcY2ka44MIjXmUM=">AAAB7XicdVDLSgMxFM3UV62vqks3waKIiyEzlrYuhIILXUkF+4B2KJk008ZmHiQZsQz9BzcuFHHrr7gWXPgJ/oWZVkFFD1w4nHMv997jRpxJhdCbkZmZnZtfyC7mlpZXVtfy6xsNGcaC0DoJeShaLpaUs4DWFVOctiJBse9y2nSHx6nfvKJCsjC4UKOIOj7uB8xjBCstNc663pGd6+YLyDyslOxiCSITobJlWymxy8WDIrS0kqJQ3X+2dt+vX2vd/EunF5LYp4EiHEvZtlCknAQLxQin41wnljTCZIj7tK1pgH0qnWRy7RjuaKUHvVDoChScqN8nEuxLOfJd3eljNZC/vVT8y2vHyqs4CQuiWNGATBd5MYcqhOnrsMcEJYqPNMFEMH0rJAMsMFE6oDSEr0/h/6RhmxYyrXOdxgmYIgu2wDbYAxYogyo4BTVQBwRcghtwB+6N0Lg1HozHaWvG+JzZBD9gPH0ArkiSRA==</latexit><latexit sha1_base64="I/gpFOl+SGFJtcY2ka44MIjXmUM=">AAAB7XicdVDLSgMxFM3UV62vqks3waKIiyEzlrYuhIILXUkF+4B2KJk008ZmHiQZsQz9BzcuFHHrr7gWXPgJ/oWZVkFFD1w4nHMv997jRpxJhdCbkZmZnZtfyC7mlpZXVtfy6xsNGcaC0DoJeShaLpaUs4DWFVOctiJBse9y2nSHx6nfvKJCsjC4UKOIOj7uB8xjBCstNc663pGd6+YLyDyslOxiCSITobJlWymxy8WDIrS0kqJQ3X+2dt+vX2vd/EunF5LYp4EiHEvZtlCknAQLxQin41wnljTCZIj7tK1pgH0qnWRy7RjuaKUHvVDoChScqN8nEuxLOfJd3eljNZC/vVT8y2vHyqs4CQuiWNGATBd5MYcqhOnrsMcEJYqPNMFEMH0rJAMsMFE6oDSEr0/h/6RhmxYyrXOdxgmYIgu2wDbYAxYogyo4BTVQBwRcghtwB+6N0Lg1HozHaWvG+JzZBD9gPH0ArkiSRA==</latexit><latexit sha1_base64="/1K3hh3FBI5sBQ080byuw0vfeW8=">AAAB7XicdVBNSwMxEM3Wr1q/qh69BIvgacmupa0HoeBBT1LB1kK7lGyabWOzyZJkhVL6H7x4UMSr/8eb/8ZsW0FFHww83pthZl6YcKYNQh9Obml5ZXUtv17Y2Nza3inu7rW0TBWhTSK5VO0Qa8qZoE3DDKftRFEch5zehqPzzL+9p0ozKW7MOKFBjAeCRYxgY6XWVS868wu9Ygm5p7WKX65A5CJU9XwvI361fFKGnlUylMACjV7xvduXJI2pMIRjrTseSkwwwcowwum00E01TTAZ4QHtWCpwTHUwmV07hUdW6cNIKlvCwJn6fWKCY63HcWg7Y2yG+reXiX95ndREtWDCRJIaKsh8UZRyaCTMXod9pigxfGwJJorZWyEZYoWJsQFlIXx9Cv8nLd/1kOtdo1L9YhFHHhyAQ3AMPFAFdXAJGqAJCLgDD+AJPDvSeXRenNd5a85ZzOyDH3DePgGmbo6F</latexit>
ms<latexit sha1_base64="fimh0WjPAqaAw4/gbE6Qd7027/k=">AAAB7XicdVA9SwNBEJ2LXzFGjVpqsRgEq+PuDEnsAhZaRjAfkBxxb7OXrNn7YHdPCEdKexsLRWz9P3b+G/cSBRV9MPB4b4aZeV7MmVSW9W7klpZXVtfy64WN4ubWdmlnty2jRBDaIhGPRNfDknIW0pZiitNuLCgOPE473uQs8zu3VEgWhVdqGlM3wKOQ+YxgpaV2MEjlrDAolS3ztF51KlVkmZZVsx07I06tclJBtlYylBsHd8VrAGgOSm/9YUSSgIaKcCxlz7Zi5aZYKEY4nRX6iaQxJhM8oj1NQxxQ6abza2foSCtD5EdCV6jQXP0+keJAymng6c4Aq7H87WXiX14vUX7dTVkYJ4qGZLHITzhSEcpeR0MmKFF8qgkmgulbERljgYnSAWUhfH2K/idtx7Qt077UaZzDAnnYh0M4Bhtq0IALaEILCNzAPTzCkxEZD8az8bJozRmfM3vwA8brB3xikIc=</latexit><latexit sha1_base64="QQJwveTIyWi+w+V9E8bmr+9O2gc=">AAAB7XicdVA9SwNBEN3zM8ao0ZRaLAbB6tiLIYldwELLCOYDkjPsbfaSNXt7x+6eEI6U9jYWitj6A+z9EXb+G/cSBRV9MPB4b4aZeV7EmdIIvVsLi0vLK6uZtex6bmNzK7+901JhLAltkpCHsuNhRTkTtKmZ5rQTSYoDj9O2Nz5J/fY1lYqF4kJPIuoGeCiYzwjWRmoF/URNs/18EdnHtUqpXIHIRqjqlJyUlKrlozJ0jJKiWN+7yV2+FF4b/fxbbxCSOKBCE46V6joo0m6CpWaE02m2FysaYTLGQ9o1VOCAKjeZXTuFB0YZQD+UpoSGM/X7RIIDpSaBZzoDrEfqt5eKf3ndWPs1N2EiijUVZL7IjznUIUxfhwMmKdF8YggmkplbIRlhiYk2AaUhfH0K/yetku0g2zk3aZyCOTJgF+yDQ+CAKqiDM9AATUDAFbgF9+DBCq0769F6mrcuWJ8zBfAD1vMHgB2SCQ==</latexit><latexit sha1_base64="QQJwveTIyWi+w+V9E8bmr+9O2gc=">AAAB7XicdVA9SwNBEN3zM8ao0ZRaLAbB6tiLIYldwELLCOYDkjPsbfaSNXt7x+6eEI6U9jYWitj6A+z9EXb+G/cSBRV9MPB4b4aZeV7EmdIIvVsLi0vLK6uZtex6bmNzK7+901JhLAltkpCHsuNhRTkTtKmZ5rQTSYoDj9O2Nz5J/fY1lYqF4kJPIuoGeCiYzwjWRmoF/URNs/18EdnHtUqpXIHIRqjqlJyUlKrlozJ0jJKiWN+7yV2+FF4b/fxbbxCSOKBCE46V6joo0m6CpWaE02m2FysaYTLGQ9o1VOCAKjeZXTuFB0YZQD+UpoSGM/X7RIIDpSaBZzoDrEfqt5eKf3ndWPs1N2EiijUVZL7IjznUIUxfhwMmKdF8YggmkplbIRlhiYk2AaUhfH0K/yetku0g2zk3aZyCOTJgF+yDQ+CAKqiDM9AATUDAFbgF9+DBCq0769F6mrcuWJ8zBfAD1vMHgB2SCQ==</latexit><latexit sha1_base64="cNBVvQQBwsEOvudlTZEsw54D8kM=">AAAB7XicdVDLSgMxFM34rPVVdekmWARXQ2Ysbd0VXOiygn1AO5RMmmljk8yQZIQy9B/cuFDErf/jzr8x01ZQ0QMXDufcy733hAln2iD04aysrq1vbBa2its7u3v7pYPDto5TRWiLxDxW3RBrypmkLcMMp91EUSxCTjvh5DL3O/dUaRbLWzNNaCDwSLKIEWys1BaDTM+Kg1IZuRf1ql+pQuQiVPN8Lyd+rXJegZ5VcpTBEs1B6b0/jEkqqDSEY617HkpMkGFlGOF0VuynmiaYTPCI9iyVWFAdZPNrZ/DUKkMYxcqWNHCufp/IsNB6KkLbKbAZ699eLv7l9VIT1YOMySQ1VJLFoijl0MQwfx0OmaLE8KklmChmb4VkjBUmxgaUh/D1KfyftH3XQ653g8qNq2UcBXAMTsAZ8EANNMA1aIIWIOAOPIAn8OzEzqPz4rwuWlec5cwR+AHn7RO6Yo86</latexit>
mud<latexit sha1_base64="0x4aBwaMF94gOwOxwplmtAl1wQ0=">AAAB7nicdVBNS8NAEJ3Ur1qrVj3qYbEInkISS1tvBQ96rGBtoQ11s9m0Szcf7G6EEnr0B3jxoIhXf483/42bVkFFHww83pthZp6XcCaVZb0bhaXlldW14nppo7y5tV3Z2b2WcSoI7ZCYx6LnYUk5i2hHMcVpLxEUhx6nXW9ylvvdWyoki6MrNU2oG+JRxAJGsNJSNxxmqT8rDStVyzxt1p1aHVmmZTVsx86J06id1JCtlRzV1sFd+QYA2sPK28CPSRrSSBGOpezbVqLcDAvFCKez0iCVNMFkgke0r2mEQyrdbH7uDB1pxUdBLHRFCs3V7xMZDqWchp7uDLEay99eLv7l9VMVNN2MRUmqaEQWi4KUIxWj/HfkM0GJ4lNNMBFM34rIGAtMlE4oD+HrU/Q/uXZM2zLtS53GOSxQhH04hGOwoQEtuIA2dIDABO7hEZ6MxHgwno2XRWvB+JzZgx8wXj8APn+Q9w==</latexit><latexit sha1_base64="PRipC2c6F/uP9hCOKTVyGDupSe8=">AAAB7nicdVBNS8NAEN3Ur1qrVnvUw2IRPIUklrbeCh70WMF+QBvLZrNpl+4mYXcjlNCjP8CLB0W8evfuj/Dmv3HTKqjog4HHezPMzPNiRqWyrHcjt7S8srqWXy9sFDe3tks7ux0ZJQKTNo5YJHoekoTRkLQVVYz0YkEQ9xjpepPTzO9eEyFpFF6qaUxcjkYhDShGSktdPkwTf1YYliqWedKoOdUatEzLqtuOnRGnXj2uQlsrGSrN/Zvi1Uv5tTUsvQ38CCechAozJGXftmLlpkgoihmZFQaJJDHCEzQifU1DxIl00/m5M3ioFR8GkdAVKjhXv0+kiEs55Z7u5EiN5W8vE//y+okKGm5KwzhRJMSLRUHCoIpg9jv0qSBYsakmCAuqb4V4jATCSieUhfD1KfyfdBzTtkz7QqdxBhbIgz1wAI6ADeqgCc5BC7QBBhNwC+7BgxEbd8aj8bRozRmfM2XwA8bzB0I6knk=</latexit><latexit sha1_base64="PRipC2c6F/uP9hCOKTVyGDupSe8=">AAAB7nicdVBNS8NAEN3Ur1qrVnvUw2IRPIUklrbeCh70WMF+QBvLZrNpl+4mYXcjlNCjP8CLB0W8evfuj/Dmv3HTKqjog4HHezPMzPNiRqWyrHcjt7S8srqWXy9sFDe3tks7ux0ZJQKTNo5YJHoekoTRkLQVVYz0YkEQ9xjpepPTzO9eEyFpFF6qaUxcjkYhDShGSktdPkwTf1YYliqWedKoOdUatEzLqtuOnRGnXj2uQlsrGSrN/Zvi1Uv5tTUsvQ38CCechAozJGXftmLlpkgoihmZFQaJJDHCEzQifU1DxIl00/m5M3ioFR8GkdAVKjhXv0+kiEs55Z7u5EiN5W8vE//y+okKGm5KwzhRJMSLRUHCoIpg9jv0qSBYsakmCAuqb4V4jATCSieUhfD1KfyfdBzTtkz7QqdxBhbIgz1wAI6ADeqgCc5BC7QBBhNwC+7BgxEbd8aj8bRozRmfM2XwA8bzB0I6knk=</latexit><latexit sha1_base64="x7NVE2FJJIQx4Ac2iCaHilpJfv8=">AAAB7nicdVDLSgMxFM3UV62vqks3wSK4GjK1tHVXcKHLCvYB7VAymUwbmmSGJCOUoR/hxoUibv0ed/6NmbaCih64cDjnXu69J0g40wahD6ewtr6xuVXcLu3s7u0flA+PujpOFaEdEvNY9QOsKWeSdgwznPYTRbEIOO0F06vc791TpVks78wsob7AY8kiRrCxUk+MsjScl0blCnIvm/VqrQ6Ri1DDq3o5qTZqFzXoWSVHBazQHpXfh2FMUkGlIRxrPfBQYvwMK8MIp/PSMNU0wWSKx3RgqcSCaj9bnDuHZ1YJYRQrW9LAhfp9IsNC65kIbKfAZqJ/e7n4lzdITdT0MyaT1FBJlouilEMTw/x3GDJFieEzSzBRzN4KyQQrTIxNKA/h61P4P+lWXQ+53i2qtK5XcRTBCTgF58ADDdACN6ANOoCAKXgAT+DZSZxH58V5XbYWnNXMMfgB5+0TfH+Pqg==</latexit>
1st order
1st order
physical point
Z(2)
Z(2)
Crossover
!/T=0
1st order tr.
1st order tr.
2nd order 3d Ising
tri critical line
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tri cr
itica
l line
mud<latexit sha1_base64="0x4aBwaMF94gOwOxwplmtAl1wQ0=">AAAB7nicdVBNS8NAEJ3Ur1qrVj3qYbEInkISS1tvBQ96rGBtoQ11s9m0Szcf7G6EEnr0B3jxoIhXf483/42bVkFFHww83pthZp6XcCaVZb0bhaXlldW14nppo7y5tV3Z2b2WcSoI7ZCYx6LnYUk5i2hHMcVpLxEUhx6nXW9ylvvdWyoki6MrNU2oG+JRxAJGsNJSNxxmqT8rDStVyzxt1p1aHVmmZTVsx86J06id1JCtlRzV1sFd+QYA2sPK28CPSRrSSBGOpezbVqLcDAvFCKez0iCVNMFkgke0r2mEQyrdbH7uDB1pxUdBLHRFCs3V7xMZDqWchp7uDLEay99eLv7l9VMVNN2MRUmqaEQWi4KUIxWj/HfkM0GJ4lNNMBFM34rIGAtMlE4oD+HrU/Q/uXZM2zLtS53GOSxQhH04hGOwoQEtuIA2dIDABO7hEZ6MxHgwno2XRWvB+JzZgx8wXj8APn+Q9w==</latexit><latexit sha1_base64="PRipC2c6F/uP9hCOKTVyGDupSe8=">AAAB7nicdVBNS8NAEN3Ur1qrVnvUw2IRPIUklrbeCh70WMF+QBvLZrNpl+4mYXcjlNCjP8CLB0W8evfuj/Dmv3HTKqjog4HHezPMzPNiRqWyrHcjt7S8srqWXy9sFDe3tks7ux0ZJQKTNo5YJHoekoTRkLQVVYz0YkEQ9xjpepPTzO9eEyFpFF6qaUxcjkYhDShGSktdPkwTf1YYliqWedKoOdUatEzLqtuOnRGnXj2uQlsrGSrN/Zvi1Uv5tTUsvQ38CCechAozJGXftmLlpkgoihmZFQaJJDHCEzQifU1DxIl00/m5M3ioFR8GkdAVKjhXv0+kiEs55Z7u5EiN5W8vE//y+okKGm5KwzhRJMSLRUHCoIpg9jv0qSBYsakmCAuqb4V4jATCSieUhfD1KfyfdBzTtkz7QqdxBhbIgz1wAI6ADeqgCc5BC7QBBhNwC+7BgxEbd8aj8bRozRmfM2XwA8bzB0I6knk=</latexit><latexit sha1_base64="PRipC2c6F/uP9hCOKTVyGDupSe8=">AAAB7nicdVBNS8NAEN3Ur1qrVnvUw2IRPIUklrbeCh70WMF+QBvLZrNpl+4mYXcjlNCjP8CLB0W8evfuj/Dmv3HTKqjog4HHezPMzPNiRqWyrHcjt7S8srqWXy9sFDe3tks7ux0ZJQKTNo5YJHoekoTRkLQVVYz0YkEQ9xjpepPTzO9eEyFpFF6qaUxcjkYhDShGSktdPkwTf1YYliqWedKoOdUatEzLqtuOnRGnXj2uQlsrGSrN/Zvi1Uv5tTUsvQ38CCechAozJGXftmLlpkgoihmZFQaJJDHCEzQifU1DxIl00/m5M3ioFR8GkdAVKjhXv0+kiEs55Z7u5EiN5W8vE//y+okKGm5KwzhRJMSLRUHCoIpg9jv0qSBYsakmCAuqb4V4jATCSieUhfD1KfyfdBzTtkz7QqdxBhbIgz1wAI6ADeqgCc5BC7QBBhNwC+7BgxEbd8aj8bRozRmfM2XwA8bzB0I6knk=</latexit><latexit sha1_base64="x7NVE2FJJIQx4Ac2iCaHilpJfv8=">AAAB7nicdVDLSgMxFM3UV62vqks3wSK4GjK1tHVXcKHLCvYB7VAymUwbmmSGJCOUoR/hxoUibv0ed/6NmbaCih64cDjnXu69J0g40wahD6ewtr6xuVXcLu3s7u0flA+PujpOFaEdEvNY9QOsKWeSdgwznPYTRbEIOO0F06vc791TpVks78wsob7AY8kiRrCxUk+MsjScl0blCnIvm/VqrQ6Ri1DDq3o5qTZqFzXoWSVHBazQHpXfh2FMUkGlIRxrPfBQYvwMK8MIp/PSMNU0wWSKx3RgqcSCaj9bnDuHZ1YJYRQrW9LAhfp9IsNC65kIbKfAZqJ/e7n4lzdITdT0MyaT1FBJlouilEMTw/x3GDJFieEzSzBRzN4KyQQrTIxNKA/h61P4P+lWXQ+53i2qtK5XcRTBCTgF58ADDdACN6ANOoCAKXgAT+DZSZxH58V5XbYWnNXMMfgB5+0TfH+Pqg==</latexit>
Nf = 2<latexit sha1_base64="lDZ6GTcUZ43Dbrzla29lS8np/dM=">AAAB7XicdVDLSgNBEOyNrxhfUY9eBoMgHpbdNSTxIAQ86EkimAckS5idzCZjZh/MzAphyT948aCIV3/Fszf/xtlEQUULGoqqbrq7vJgzqSzr3cgtLC4tr+RXC2vrG5tbxe2dlowSQWiTRDwSHQ9LyllIm4opTjuxoDjwOG1747PMb99SIVkUXqtJTN0AD0PmM4KVllqXff/UKfSLJcs8qVWccgVZpmVVbcfOiFMtH5eRrZUMpfrRq40AoNEvvvUGEUkCGirCsZRd24qVm2KhGOF0WuglksaYjPGQdjUNcUClm86unaIDrQyQHwldoUIz9ftEigMpJ4GnOwOsRvK3l4l/ed1E+TU3ZWGcKBqS+SI/4UhFKHsdDZigRPGJJpgIpm9FZIQFJkoHlIXw9Sn6n7Qc07ZM+0qncQ5z5GEP9uEQbKhCHS6gAU0gcAN38ACPRmTcG0/G87w1Z3zO7MIPGC8fsdOQCA==</latexit><latexit sha1_base64="I/gpFOl+SGFJtcY2ka44MIjXmUM=">AAAB7XicdVDLSgMxFM3UV62vqks3waKIiyEzlrYuhIILXUkF+4B2KJk008ZmHiQZsQz9BzcuFHHrr7gWXPgJ/oWZVkFFD1w4nHMv997jRpxJhdCbkZmZnZtfyC7mlpZXVtfy6xsNGcaC0DoJeShaLpaUs4DWFVOctiJBse9y2nSHx6nfvKJCsjC4UKOIOj7uB8xjBCstNc663pGd6+YLyDyslOxiCSITobJlWymxy8WDIrS0kqJQ3X+2dt+vX2vd/EunF5LYp4EiHEvZtlCknAQLxQin41wnljTCZIj7tK1pgH0qnWRy7RjuaKUHvVDoChScqN8nEuxLOfJd3eljNZC/vVT8y2vHyqs4CQuiWNGATBd5MYcqhOnrsMcEJYqPNMFEMH0rJAMsMFE6oDSEr0/h/6RhmxYyrXOdxgmYIgu2wDbYAxYogyo4BTVQBwRcghtwB+6N0Lg1HozHaWvG+JzZBD9gPH0ArkiSRA==</latexit><latexit sha1_base64="I/gpFOl+SGFJtcY2ka44MIjXmUM=">AAAB7XicdVDLSgMxFM3UV62vqks3waKIiyEzlrYuhIILXUkF+4B2KJk008ZmHiQZsQz9BzcuFHHrr7gWXPgJ/oWZVkFFD1w4nHMv997jRpxJhdCbkZmZnZtfyC7mlpZXVtfy6xsNGcaC0DoJeShaLpaUs4DWFVOctiJBse9y2nSHx6nfvKJCsjC4UKOIOj7uB8xjBCstNc663pGd6+YLyDyslOxiCSITobJlWymxy8WDIrS0kqJQ3X+2dt+vX2vd/EunF5LYp4EiHEvZtlCknAQLxQin41wnljTCZIj7tK1pgH0qnWRy7RjuaKUHvVDoChScqN8nEuxLOfJd3eljNZC/vVT8y2vHyqs4CQuiWNGATBd5MYcqhOnrsMcEJYqPNMFEMH0rJAMsMFE6oDSEr0/h/6RhmxYyrXOdxgmYIgu2wDbYAxYogyo4BTVQBwRcghtwB+6N0Lg1HozHaWvG+JzZBD9gPH0ArkiSRA==</latexit><latexit sha1_base64="/1K3hh3FBI5sBQ080byuw0vfeW8=">AAAB7XicdVBNSwMxEM3Wr1q/qh69BIvgacmupa0HoeBBT1LB1kK7lGyabWOzyZJkhVL6H7x4UMSr/8eb/8ZsW0FFHww83pthZl6YcKYNQh9Obml5ZXUtv17Y2Nza3inu7rW0TBWhTSK5VO0Qa8qZoE3DDKftRFEch5zehqPzzL+9p0ozKW7MOKFBjAeCRYxgY6XWVS868wu9Ygm5p7WKX65A5CJU9XwvI361fFKGnlUylMACjV7xvduXJI2pMIRjrTseSkwwwcowwum00E01TTAZ4QHtWCpwTHUwmV07hUdW6cNIKlvCwJn6fWKCY63HcWg7Y2yG+reXiX95ndREtWDCRJIaKsh8UZRyaCTMXod9pigxfGwJJorZWyEZYoWJsQFlIXx9Cv8nLd/1kOtdo1L9YhFHHhyAQ3AMPFAFdXAJGqAJCLgDD+AJPDvSeXRenNd5a85ZzOyDH3DePgGmbo6F</latexit>
Nf = 3
!/T=i"/3
mud
ms
PureGauge
1st
1st
crossover
Nf=
3phys.point
Z(2)
Z(2)
Nf = 2
mtrics
Nf
=1
O(4)?U(2)L ¢ U(2)R/U(2)V ?
chiral limit Nf = 2
Columbia plot in 𝜇=0 and RW plane
!20
𝜇/T=𝜋/3
𝜇/T=0