Finite Density Lattice QCDy Qby a Histogram Method
Shinji EjiriShinji EjiriNiigata University
WHOT-QCD collaborationWHOT QCD collaboration S. Ejiri1, S. Aoki2, T. Hatsuda3,4, K. Kanaya2, Y. Maezawa5,
Y. Nakagawa1, H. Ohno2,6, H. Saito2, and T. Umeda7
1Niigata Univ., 2Univ. of Tsukuba, 3Univ. of Tokyo, 4RIKEN, 5BNL, 6Bielefeld Univ., 7Hiroshima Univ.
Workshop on Quarks and Hadrons under Extreme Conditions, Keio, November 17-18, 2011
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
Problems in simulations at 0 Problem of Complex Determinant at 0
Boltzmann weight: complex at 0 Configurations cannot be generated. Monte-Carlo method is not applicable.
i f h d ( i h d) Density of state method (Histogram method)X: order parameters, total quark number, average plaquette etc.
,,,,, TmXWdXTmZhistogram
SN Expectation values
gSNmMXX-DUTmXW e,det ,,, f
,,, 1,,
TmXWXOdX
ZXO
Tm
W(X) can be defined even when detM is complex. Sign problem
,, Z
Sign problem
gSN emMXXDUXW ,det' ' f
gSNi emMeXXDU ,det ' f
generate the configuration: complex phase of detM
generate the configuration with this weight
• Sign problem: If changes its sign frequently,ie
error) al(statistic ~)(fixed
X
ieXW
Overlap problem lnexp1 1
eff dXXOXVZ
dXXWXOZ
XOZZ)(ln)(eff XWXV
• W is computed from the histogram.• Distribution function around X where XOXV ln)(eff
is minimized: important.• Veff must be computed in a wide range.
)(eff
p g
XOXV ln)(eff XOln)(eff XV
+ = Out of the reliable range
reliable range
reliable rangeIf X-dependence is large. reliable range
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=0
• Pressure at 0 0
3
344 )0(det)(detln
)0()(ln10
MM
NN
ZZ
VTTp
Tp
s
t
• Calculation of and : required.
P )0(det)(det MM
Distribution function for Equation of state gSN emMFFPPDUmTFPW ,det'' ,,,',' f
gSN emMPPDUmTPW ,det' ,,,' for
,,,, ,, TmFPWdPdFTmZ
)0(det)(detlnf MMNF
,6 site PNSg
• We propose a method for the calculation of this W.
,)0(det)(detlnf MMNF
– We must calculate W in a wide range of P and F.– The sign problem must be solved to obtain W.
O h l l• Once we get the pressure, we can calculate
4TPn 42 TPq C iti l i t h
,3 TT
23 TT
q
etc. Critical point search
-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
Double well potential
Plan of this talk
• Introduction: Distribution function• Test in the heavy quark region• Application to the light quark region at finite densityApplication to the light quark region at finite density
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.
(, 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
Solving the overlap problemEffective 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 KMKM
Isospin chemical potential(Nf=2), u= d.
KMdet
,det,det,det 2 KMKMKM
IR
Ns
N TiTKNPKNM
KMtt sinhcosh212288
0,0det,detln 34
sitephase
• If the complex phase is neglected,
TKK tt NN cosh cpK
– Critical point:
TKK cosh
0cosh cpcptt NN KTK
TKK tt NN cosh0
• Even if we add the effect from the complex
TKK tt cosh0cpcp
phase, it is small for the determination of cp. (WHOT-QCD Collab., in preparation, 11) TK tN sinh
Distribution function in the light quark regionWHOT-QCD Collaboration in preparation (arXiv:1111 2116)WHOT-QCD Collaboration, in preparation, (arXiv:1111.2116)
• We perform phase quenched simulationsp p q• The effect of the complex phase is added by the
reweightingreweighting.• We calculate the probability distribution function.
• GoalGoal– The critical point
Th ti f t t– The equation of statePressure, Energy density, Quark number density, Quark number
susceptibility Speed of sound etcsusceptibility, 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
• 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().• Tail of the distribution W0 is important. 0ie
– Computed by simulations out of the pion condensed phase.
Th l l ti f W i id f
pion condensed phase
/2• The calculations of W0 in a wide range of (P,F) is very important.
m/2
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.
Distribution of the complex phase
• 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),
CFP
ie 2
, 21exp
W WW
2No information
C
2
Complex phase G i di t ib ti Th l t i i d• Gaussian distribution The cumulant expansion is good.
• We define the phase
Td
TMN
MMN
T
0ffdetlnIm
0detdetlnIm
– The range of is from - to .
• At the same time, we calculate F as a function of ,
Td
TMN
MMNF
T
0ffdetlnRe
0detdetln
• The reweighting factor is also computed,
Td
TMN
MMNC
T
T0
detlnRedetdetln f
0f
fixed ,000
0f
detdet
,,,,,
FP
N
MM
FPWFPWFPR
F and C are strongly correlated. R: small error
Integral method for the calculation of the quark determinantTT
TM
detln
TT
T4.0 T
lattice 483 50.1 TT
8.0 mm
42T
T T
2-flavor QCDIwasaki gauge + clover Wison
4.2 T
quark action
Random noise TT
Real part Imaginary partmethod is used.TT
Distribution of the complex phase
4.07.1
T40
5.1
T 4.0 T
4.27.1
T4.2
5.1
T
• Well approximated by a Gaussian function.Well approximated by a Gaussian function.• Convergence of the cumulant expansion: good.
Overlap problem
• 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
Convergence of the cumulant expansiond thF-dependence of the 2nd and 4th order cumulants
of the complex phasePreliminaryPreliminary
Preliminary• 4th order cumulant is consistent with zero.
Preliminary
Convergence of the cumulant expansiond thP-dependence of the 2nd and 4th order cumulants
of the complex phase
Preliminary• 4th order cumulant is consistent with zero.
Preliminary
Effective potential at finite • Combine V0 and the phase factor. Preliminary
FPWln 2 FPW ,ln 0C
2
+ ×0.5
FPW ,ln
=
Summary• We studied the quark mass and chemical potential dependence
of QCD phase transition in the heavy quark region.Q p y q g
• The histogram method is used to investigte the nature of the g gphase transition.
• To avoid the sign problem, the method based on the cumulant expansion of is useful.
• Further studies in light quark region are important applying this method.
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