finite element analysis of a bus body structure using cae tools
Phase structure of finite densityPhase structure of finite density … · 2012. 2. 24. · Phase...
Transcript of Phase structure of finite densityPhase structure of finite density … · 2012. 2. 24. · Phase...
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Phase structure of finite densityPhase structure of finite density lattice QCD by a histogram methodQ y g
Shinji EjiriShinji EjiriNiigata University
WHOT-QCD collaboration S. Ejiri1, S. Aoki2, T. Hatsuda3,4, K. Kanaya2,
Y. Nakagawa1, H. Ohno2,5, H. Saito2, and T. Umeda6
1Niigata Univ 2Univ of Tsukuba 3Univ of Tokyo 4RIKEN
YIPQS HPCI i i l l l k h N f
1Niigata Univ., 2Univ. of Tsukuba, 3Univ. of Tokyo, 4RIKEN, 5Bielefeld Univ., 6Hiroshima Univ.
YIPQS-HPCI international molecule-type workshop on New-type of Fermions on the Lattice (YITP, Kyoto, Feb.9-24, 2012)
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Phase structure of QCD at high temperature and density
Lattice QCD Simulations quark-gluon plasma phase
• Phase transition lines• Equation of state
RH
I SP
T
LH
• Direct simulation:
C PS
A
RHIC low-EFAIR
C
• Direct simulation: Impossible at 0.
AG
S deconfinement?
hadron phase color super
quarkyonic?chiral SB?
phase color super conductor?nuclear
matterq
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Probability distribution function Distribution function (Histogram)
X: order parameters, total quark number, average plaquette etc.
,,, ,, TmXWdXTmZhistogram
In the Matsubara formalism,
SN
gSNmMDUTmZ e ,det,, f
h d M k d i S i
gSNmMXX-DUTmXW e,det,,, f
where detM: quark determinant, Sg: gauge action.
Useful to identify the nature of phase transitions Useful to identify the nature of phase transitions e.g. At a first order transition, two peaks are expected in W(X).
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-dependence of the effective potential )(ln)(eff XWXV ,,,, TXWdXTZ
X: order parameters, total quark number, average plaquette, quark determinant etc.
Crossover
Critical point ,,eff TXV Correlation length: short V(X): Quadratic function
Correlation length: long
T
Correlation length: longCurvature: Zero
TQGP
1st order phase transition
hadron CSC?
1 order phase transition
Two phases coexistD bl ll i l
CSC? Double well potential
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Quark mass dependence of the critical point
2ndorder 1storder
QuenchedNf=2
Physical point
2ndorder 1storder y p
Crossoverms
1storderCrossover
0
mud0
0
• Where is the physical point?• Extrapolation to finite density
– investigating the quark mass dependence near =0• Critical point at finite density?
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Equation of State• Integral method
– Interaction measure ln 13 Zp
e ac o easu e
computed by plaquette (1x1 Wilson loop) and the derivative of detM.
,ln34 aVTT
P
– Pressure at =0Z
VTTp ln1
34
• Integral
a
adT
pTp
Tp ln3
444
VTT
aaa TTT 0
0
444
a0: start point p=03
)(d t)(1 MNZ• Pressure at 0, 0
344 )0(det)(detln
)0()(ln10
MM
NN
ZZ
VTTp
Tp
s
t
• with )0(det)(detor MMPX ,,, 1,,
TmXWXdX
ZX
Tm
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Plan of this talk
• Test in the heavy quark region– H. Saito et al. (WHOT-QCD Collab.), Phys.Rev.D84, 054502(2011)– WHOT-QCD Collaboration, in preparation
• Application to the light quark region at finite densityApplication to the light quark region at finite density– S.E., Phys.Rev.D77, 014508(2008))– WHOT-QCD Collaboration, in preparationWHOT QCD Collaboration, in preparation
(Lattice 2011 proc.: Y. Nakagawa et al., arXiv:1111.2116)
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Distribution function in the heavy quark regionWHOT QCD C ll b Ph R D84 054502(2011)WHOT-QCD Collab., Phys.Rev.D84, 054502(2011)
• We study the critical• We study the critical surface in the (mud, ms, ) space in the heavy quarkspace in the heavy quark region.
• Performing quenched• Performing quenched simulations + Reweighting.Reweighting.– plaquette gauge action +
Wilson quark action
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(, m, )-dependence of the Distribution function• Distributions of plaquette P (1x1 Wilson loop for the standard action)
PNNmMPP-DUmPW sitef 6e,det,,, mMPPDUmPW e,det,,, (Reweight factor 0,,,,,,,,,, 0000 mPWmPWmmPR
'6)0(0
'6f
f
det0,det,det
NPN
N
PN mMmMmMPP-
'
0
'6
)0,(
)0,('6 0site
0
00site
0,det,det
P
PNPN
mMmMe
PP-ePR
Effective potential:
,,,,ln0,,,,,ln,,, 0000ffff mmPRmPVmPWmPV ,,,,ln0,,,,,ln,,, 0000effeff mmPRmPVmPWmPV
NmMPNPR
f,detl6l
PmM
PNPR0,det
, ln6ln0
0site
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Distribution function in quenched simulationsEffective potential in a wide range of P: requiredEffective potential in a wide range of P: required.
Plaquette histogram at K=1/mq=0. Derivative of Veff at =5.69
dVeff/dP is adjusted to =5.69, using 12site1eff
2eff 6 N
dPdV
dPdV
,4243site N 5 points, quenched.
j , g
These data are combined by taking the average.
12site12 dPdP
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Effective potential near the quenched limitWHOT QCD Phys Rev D84 054502(2011)WHOT-QCD, Phys.Rev.D84, 054502(2011)
Quenched Simulation(mq=, K=0)
first order
dPdVeff
K~1/mq for large mq
dP
crossover
Quark masssmaller
• detM: Hopping parameter expansionlattice, 4243 5 points, Nf=2
• detM: Hopping parameter expansion,Nf=2: Kcp=0.0658(3)(8)
020T
R
Ns
N tt KNPKNNM
KMN 34siteff 212288
0detdetln
l t f P l k l
• First order transition at K = 0 changes to crossover at K > 0.02.0
mTc
real part of Polyakov loop
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Endpoint of 1st order transition in 2+1 flavor QCD
KMd
Nf=2: Kcp=0.0658(3)(8)
R
Ns
N tt KNPKNM
KM 34site 2122882
0detdetln2
Nf=2+1
NNN
sud
MKMKM
0detdetdetln
344
3
2
RNs
Nuds
Nsudsite
ttt KKNPKKN 22122288 344
The critical line is described by
t22 NNN KKK tt t2)fcp(22 Nsud KKK tt
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Finite density QCD in the heavy quark region
xUxUxUxU aa †4
†444
qq e ,e in detM
KMdet
** qq e , TTe Polyakov loop
NN
TTNs
N
TiTKNPKNN
eeKNPKNNM
KMN
tt
tt
i hh212288
262880,0det,detln
34
*//34siteff
IRN
sN TiTKNPKNN tt sinhcosh212288 34
sitefphase
iPolyakov loop
IR idistribution
• We can extend this discussion to finite density QCD.to finite density QCD.
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Phase quenched simulations, Isospin chemical potential
*,det,det KMKMIsospin chemical potential
(Nf=2), u= d. ,det,det,det 2 KMKMKM
IR
Ns
N TiTKNPKNM
KMtt sinhcosh212288
0,0det,detln 34
sitephase
• If the complex phase is neglected,
M 0,0det phase
Nf=2
TKK tt NN cosh Nf=2
– Critical point:
NN 0cosh cpcptt NN KTK
tN TKK cosh0cpcp
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Distribution function for P and R at =0
gSNRRR eMPPDUPW det'' ,,',' f
W
NN
PRN
sN
tt
R
tt
KNNPNKN
KNNPKNNPNWW
34
fixed ,
3f
4sitefsite0
0
2122886exp
2122886exp0,
,
Rs KNNPNKN fsitef0 2122886exp
RN
sN tt KNNPNKNVV 3
fsite4
f00effeff 21228860,,
• Peak position of W: 0effeff
Rd
dVdP
dV
Lines of zero derivativesfor first order
crossover1 intersection
first order transition3 intersections
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Derivatives of Veff in terms of P and R
l i4243 iIn heavy quark region, lattice,4243 5 points
RN
sN tt KNNPNKNVV 3
fsite4
f00effeff 21228860,,
site4
f00effeff 28860,, NKN
dPdV
dPKdV
tt Ns
N
RR
KNNd
dVd
KdV 3f
0effeff 2120,,
hif
Rsfsitef00effeff
dPdP RR dd constant shift constant shift
• Contour lines of and at (,) = (0,0) correspond to dPdVeff
ddV
eff (, ) ( , ) p
the lines of the zero derivatives at ().dP Rd
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Lines of constant derivatives obtained by quenched simulations
At the critical point,d /ddVeff/dP=0dVeff/dR=0
RR
Bl li dV /dPBlue lines: dVeff/dPRed lines: dVeff/dRP
• The number of the intersection changes around =0.658
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Finite density QCD in the heavy quark region
xUxUxUxU aa †4
†444
qq e ,e
KMdet
** qq e , TTe
NN
TTNs
N
TiTKNPKNN
eeKNPKNNM
KMN
tt
tt
i hh212288
262880,0det,detln
34
*//34siteff
IRN
sN TiTKNPKNN tt sinhcosh212288 34
sitefphase
• We can extend this discussion to finite density QCD.
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Finite density QCD
TTNs
N eeKNPKNNM
KMN tt262880det
detln *//34siteff
IRN
sN TiTKNPKNN tt sinhcosh212288 34
sitefphase
fixed ,
3f
4sitefsite0
0
sinhcosh2122886exp0,
,R
tt
PIRN
sN TiTKNNPKNNPN
WW
fixed ,
3fsite
4f0 cosh2122886exp
R
tt
P
iR
Ns
N eTKNNPNKN
INN TKNN tt sinh212 3
f
fixed
3fsite
4f00effeff lncosh21228860,, tt
P
iR
Ns
N eTKNNPNKNVV
Is TKNN sinh212 f
• Reweighting: is a non-linear term in
fixed , RP
.0,, 0ffff VVln ie Reweighting: is a non linear term in .0,, 0effeff VVfixed ,
lnRP
e
Sign problem.
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Avoiding the sign problem(SE, Phys.Rev.D77,014508(2008), WHOT-QCD, Phys.Rev.D82, 014508(2010))
: complex phase IN
sN TKNNM tt sinh212detlnIm 3
f
• Sign problem: If changes its sign,
error)al(statisticie
ie
• Cumulant expansion
error)al(statistic fixed ,
RP
e
<..>F,P: expectation values fixed F and P.p
CCCCP
i iieR
432
, !41
!321exp
R, !4!32
43233222 23
cumulants0 0
– Odd terms vanish from a symmetry under
CPPFPCPPCPC RRRRRRR
4,,,
2
,
33,,
22,
,23 , ,
y y Source of the complex phase
If the cumulant expansion converges, No sign problem.
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Cumulant expansion of the complex phase • When the distribution of is perfectly Gaussian, the average of
the complex phase is give by the second order (variance),Distribution of at K=0Distribution of at K=0
CFP
ie 2
, 21exp
B W( ) i h d b th f t
IN
sN TKNN tt sinh212 3
f
• Because W() is enhanced by the factor,
2 R
Ns
N TKNNW tt cosh212exp~, 3f
the region of large R is important.• The distribution of seems to be of Gaussian at R>0.
cI
• The higher order cumulants etc. might be small.cI4
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Convergence in the large volume (V) limit• Because ~ O(V), Naïve expectation:
– If so, the cumulant expansion does not converge.?)(~ n
C
n VO
However, this problem is solved in the following situation.
• The phase is given by 1 2 3The phase is given by – No correlation between x.– In the heavy quark region, the phase is the imaginary
x
x 1 2 3
4 5 6e eavy qua eg o , e p ase s e ag a ypart of the Polyakov loop average, I.
4 5 6
87 987 9
correlation length I
Ns
N TKNN tt sinh212 3f
– If the spatial correlation length is short, the distribution of the imaginary part of the Polyakov loop is expected to be Gaussian by th t l li it ththe central limit theorem.
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Convergence in the large volume (V) limitThis problem is solved in the following situation.• The phase is given by xp g y
– No correlation between x.
xx
ni i
x nC
nx
n
xPF
i
PF
i
PF
i
nieee xx
x
!exp
,,
,
nC
nn
PF
i
nie
!exp
,)(~ VO
xC
nxC
n
– Ratios of cumulants do not change in the large V limit.– Convergence property is independent of Vg p p y palthough the phase fluctuation becomes larger as V increases.
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Effect from the complex phase
I2 • The complex phase fluctuation is
l d 0 d <0cI large around =0 and <0.• However, the region around =0
is less important for finite
Id 2
is less important for finite .
dP
dRPI
,
2
R
PI
d
dR
,
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Critical surface at finite densityC t l t f th d i ti f V (P ) TK tN i h• Contour plot of the derivatives of Veff (P,R). TK tN sinh
5100.1 0
• Blue line is the dVeff/dR around the critical point• Blue line is the dVeff/dR around the critical point.• Effect from the complex phase is small on the critical line, .102 5
NNdVKdV Nt 2232230 P
NNNKNdP
dVdP
KdV cIsNt
3f
2
site4
f00effeff
22328860,,
IN NNdVKdV t 2232230
R
cIsNs
N
RR
NNT
KNNd
dVd
KdV ttt
f3
f0effeff
223cosh2120,,
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Critical line in finite density QCD(WHOT QCD Collab in preparation 11)(WHOT-QCD Collab., in preparation, 11)
• The effect from the complex IN
fsN TKNiN tt sinh212 3
phase is very small for the determination of cp because
Ifs
Critical line for u=d=
is small. • On the critical line,
TKK tt NN cosh0 cpcp
, I
Nfs
N TKNN tt tanh0212 cp3
< 1
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Distribution function in the light quark regionWHOT-QCD Collaboration, in preparation,WHOT QCD Collaboration, in preparation,
(Nakagawa et al., arXiv:1111.2116)
W f h h d i l ti• We perform phase quenched simulations• The effect of the complex phase is added by the
reweighting.• We calculate the probability distribution functionWe calculate the probability distribution function.
• Goal– The critical pointp– The equation of state
Pressure, Energy density, Quark number density, Quark number , gy y, Q y, Qsusceptibility, Speed of sound, etc.
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Probability distribution function by phase quenched simulationby phase quenched simulation
• We perform phase quenched simulations with the weight:We perform phase quenched simulations with the weight: gSN emM ,det f
,det'' ,,,',' f emMFFPPDUmFPW
SNi
SN g
''
,det'' f
mFPWe
emMeFFPPDUi
SNi g
,,,, 0','mFPWe
FP
det M expectation value with fixed P,F histogram
0det
detlnf MMNF
MN detlnImfP: plaquette
PNN eMFFPPDUFPW sitef 60 det'' ','Distribution function
of the phase quenched.
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Phase quenched simulation ,,,,,,,, 0,
mFPWemFPWFP
i
• When = d pion condensation occurs
,det,det,det 2 KMKMKM ,,det,det * KMKM
• When u=d, pion condensation occurs.• is suggested in the pion
d d h b h l i l
Phase structure of the phase quenched
2-flavor QCD0ie
condensed phase by phenomenological studies. [Han-Stephanov ’08, Sakai et al. ‘10]
N l b t W( ) d W ( )
2 flavor QCD
No overlap between W() and W0().
• Near the phase boundary, 0ieNear the phase boundary,– large fluctuations in : expected. pion condensed
phase /2 ln0 ii ee
– W(P,F) and W0(P,F) are completely different.m/2 .ln 0
,,
FPFPee
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Peak position of W(P,F) ),(ln FPW
• The slopes are zero at the peak of W(P,F).
0ln ,0ln
FW
PW
P
eFP
PWFP
PW FP
i
,0ln
,,,ln,,,ln
00
00 ,,,
,,,,,,FPWFPWFPR
P
eFP
PRNFP
PW FP
i
site
,0000
0ln
,,,ln6,,,ln =0
F
eFP
FWFP
FW FP
i
,0ln
,,,ln,,,ln If these terms are canceled
=0
F
eFP
FRFP
FW
FFF
FP
i
,000
0ln
,,,ln,,,ln
are canceled,
0 )const.(,,,,,, 000 FPWFPW
=0
• W() can be computed by simulations around (0,0).
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Avoiding the sign problem(SE, Phys.Rev.D77,014508(2008), WHOT-QCD, Phys.Rev.D82, 014508(2010))
: complex phase MdetlnIm
• Sign problem: If changes its sign,
error)al(statisticie
ie
• Cumulant expansion
error)al(statistic fixed ,
FP
e
<..>P,F: expectation values fixed F and P.p
CCCCFP
i iie 432
, !41
!321exp
, !4!32
43233222 23
cumulants0 0
– Odd terms vanish from a symmetry under
CFPFPFPFPCFPFPCFPC
43
,,,
2
,
332
,,
22,
,23 , ,
y y Source of the complex phase
If the cumulant expansion converges, No sign problem.
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Complex phase distribution h ld d fi h l h i h f• We should not define the complex phase in the range from to
• When the distribution of is perfectly Gaussian, the average of the complex phase is give by the second order (variance) 1phase is give by the second order (variance),
CFP
ie 2
, 21exp
W W
C
2No information
• Gaussian distribution The cumulant expansion is good.
p g• We define the phase
dMNMNT detlnImdetlnIm
– The range of is from - to .
Td
TN
MN
0ff Im0det
lnIm
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Distribution of the complex phase
4.07.1
T40
5.1
T 3
4.0 Tlattice483
8.0 mm2-flavor QCDIwasaki gauge
+ clover Wilson
4.27.1
T4.2
5.1
T
quark action
Random noiseRandom noise method is used.
• Well approximated by a Gaussian function.Well approximated by a Gaussian function.• Convergence of the cumulant expansion: good.
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Distribution in a wide range
• Perform phase quenched fdetN
MFPW
Reweighting method W0: distribution function in phase quenched simulations.
p qsimulations at several points.– Range of F is different.
Ch b i h i
fixed ,000
0
detdet
,,,,,
FPMM
FPWFPWFPR
• Change by reweighting method.
• Combine the data• Combine the data.
Distribution in a wide range:Distribution in a wide range:obtained.
• The error of R is small because F is fixed.
,,ln,,,ln,,ln 0000 FPWFPRFPW
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Summary• We studied the quark mass and chemical potential dependence
of the nature of QCD phase transition.Q p
• The shape of the probability distribution function changes as a p p y gfunction of the quark mass and chemical potential.
• To avoid the sign problem, the method based on the cumulant expansion of is useful.
• To find the critical point at finite density, further studies in light quark region are important applying this method.