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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)
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
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).
-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
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?
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
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)
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
(, 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
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
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
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
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.
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
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
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
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
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.
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.
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.
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
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.
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.
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
,
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,,
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
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.
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.
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
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).
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.
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
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.
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
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.
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