a method of finding the critical point in finite density qcd
DESCRIPTION
A method of finding the critical point in finite density QCD. Existence of the critical point in finite density lattice QCD Physical Review D77 (2008) 014508 [arXiv:0706.3549] Canonical partition function and finite density phase transition in lattice QCD - PowerPoint PPT PresentationTRANSCRIPT
A method of finding the critical point in finite density QCD
Shinji Ejiri(Brookhaven National Laboratory)
Existence of the critical point in finite density lattice QCDPhysical Review D77 (2008) 014508 [arXiv:0706.3549]
Canonical partition function and finite density phase transition in lattice QCDPhysical Review D 78 (2008) 074507[arXiv:0804.3227]
Tools for finite density QCD, Bielefeld, November 19-21, 2008
QCD thermodynamics at 0
• Interesting properties of QCDMeasurable in heavy-ion collisions
Critical point at finite density
• Location of the critical point ?
– Distribution function of plaquette value
– Distribution function of quark number density
• Simulations: – Bielefeld-Swansea Collab., PRD71,054508(2005).
– 2-flavor p4-improved staggered quarks with m770MeV
– 163x4 lattice
– ln det M: Taylor expansion up to O(6)
quark-gluon plasma phase
hadron phase
RH
ICS
PS
AG
S
color super conductor?
nuclear matter
q
T
Effective potential of plaquette
Physical Review D77 (2008) 014508 [arXiv:0706.3549]
Effective potential of plaquette Veff(P) Plaquette distribution function (histogram)
• First order phase transition Two phases coexists at Tc
e.g. SU(3) Pure gauge theory
• Gauge action• Partition function
,, , PWdPZ
PNS siteg 6
PWPV ln)(eff
'e det ,' f PPMDUPW gSN Effective potential
SU(3) Pure gauge theoryQCDPAX, PRD46, 4657 (1992)
histogram
histogram
)6( 2g
Problem of complex quark determinant at 0
Problem of Complex Determinant at 0
Boltzmann weight: complex at 0 Monte-Carlo method is not applicable. Configurations cannot be generated.
55 )()( MM (5-conjugate)
)(det)(det)(det * MMM
)0,(
)0,(
f
f
f 0detdet
0det
det ,
PP-
MM
PP-
MPP-DU
MPP-DUPR
N
N
N
Distribution function and Effective potential at 0 (S.E., Phys.Rev.D77, 014508(2008))
• Distributions of plaquette P (1x1 Wilson loop for the standard action)
, , PWPRdPZ PNS siteg 6
gSNMPP-DUPW e 0det , f
Effective potential:
(Weight factor at
R(P,): independent of , R(P,) can be measured at any .
, ,ln)(eff PWPRPV
1st order phase transition?
,ln PR ,ln PW
(Reweight factor
+ = ??
=0 crossovernon-singular
Reweighting for and curvature of –lnW(P)
Change: 1(T) 2(T)
Weight:
Potential:
PNSS gg 12site12 6)()(
12112 WeWW gg SS
+ =
212site1 ln6ln WPNW
Curvature of –lnW(P) does not change.
Curvature of –lnW(P,) : independent of .
, , PWPRdPZ 'e 0det ,' f PPMDUPW gSN
-dependence of the effective potential
T
hadron
QGP
CSC
+ =
1st order phase transition
,ln PR ,ln PW
Critical point
,ln PR ,ln PW ,ln PW
+ =
Crossover
=0 reweighting
=0 reweighting
Curvature: Zero
Curvature: Negative
Sign problem and phase fluctuations• Complex phase of detM
– Taylor expansion: odd terms of ln det M (Bielefeld-Swansea, PRD66, 014507 (2002))
• || > /2: Sign problem happens. changes its sign.
• Gaussian distribution– Results for p4-improved staggered
– Taylor expansion up to O(5)
– Dashed line: fit by a Gaussian function
Well approximated
CTT histogram of
)(detlnImf MN
: NOT in the range of [-, ]
T
M
TT
M
TT
M
TN
5
55
3
33
f d
detlnd
!5
1
d
detlnd
!3
1
d
detlndIm
ie
error) al(statistic )0(det
)(det
FeM
M i
N f
FeFe Fi 22
Effective potential at (S.E., Phys.Rev.D77, 014508(2008))
Results of Nf=2 p4-staggared, m/m0.7
[data in PRD71,054508(2005)]
• detM: Taylor expansion up to O(6)
• The peak position of W(P) moves left as increases at =0.
,ln,ln,,eff PRPWPV
Rln
Wlnat =0
Solid lines: reweighting factor at finite /T, R(P,)
Dashed lines: reweighting factor without complex phase factor.
Truncation error of Taylor expansion
• Solid line: O(4)
• Dashed line: O(6)
• The effect from 5th and 6th order term is small for q/T 2.5.
N
nn
n
T
M
TnNMN
0
n
ffd
detlnd
!
1)(detln
Curvature of the effective potential
• First order transition for q/T 2.5• Existence of the critical point: suggested
– Quark mass dependence: large– Study near the physical point is important.
0
,ln, ln,,2
2
2
2
2eff
2
dP
PRd
dP
PWd
dP
PVd
at q=0
Critical point:
2
2 ln
dP
Wd
+ = ?
Nf=2 p4-staggared, m/m0.7
Canonical approach
Physical Review D 78 (2008) 074507[arXiv:0804.3227]
Canonical approach• Canonical partition function
• Effective potential as a function of the quark number N.
• At the minimum,
• First order phase transition: Two phases coexist.
NN
CGC NWTNNTZTZ exp ,,
TNNTZNWNV C ),(ln)(ln)(eff
0),(ln)(ln)(eff
TN
NTZ
N
NW
N
NV C
NN
N
NZC
)(ln
)(eff NV
T
First order phase transition line
• Mixed state First order transition
• Inverse Laplace transformation by Glasgow method Kratochvila, de Forcrand, PoS (LAT2005) 167 (2005)
Nf=4 staggered fermions, lattice– Nf=4: First order for all .
In the thermodynamic limit, ,0)(eff
N
NV
N
NTZ
TC
),(ln*
cpTT
cpTT
463
3
*
sNTT
• Fugacity expansion (Laplace transformation)
canonical partition function
• Inverse Laplace transformation
– Note: periodicity
• Derivative of lnZ
Canonical partition function
N
CGC TNNTZTZ exp ,, VN /
IGCTiTN
IC iTZeTdNTZ I
0
3
3,
2
3, 0
0
f
f
)0(det
)(detdet
0
1
0)(
NSN
GCGC
GC
M
MeMDU
ZZ
Z g
R
I
0
Integral
N
NTZ
TC
),(ln*
,32, TZiTTZ GCGC
Arbitrary 0
Integral path, e.g. 1, imaginary axis2, Saddle point
• Inverse Laplace transformation
• Saddle point approximation (valid for large V, 1/V expansion)– Taylor expansion at the saddle point.
• At low density: The saddle point and the Taylor expansion coefficients can be estimated from data of Taylor expansion around =0.
Saddle point approximation(S.E., arXiv:0804.3227)
00 zT
f
I
I
N
ITiTNI
GC
IGCTiTN
IC
M
iMeTd
Z
iTZeTdNTZ
0det
det
2
)0(3
, 2
3,
03
3
0
3
3
0
0
0detln
0
f
zT
T
M
V
NSaddle point: 0z
R
I
0
Integral
Saddle point
0
tf0
n
ffd
detlnd
!
1)(detln
n
n
nn
n
n
TDNVN
T
M
TnNMN
3sNV
VN /
• Canonical partition function in a saddle point approximation
• Chemical potential
Saddle point approximation
)0,(
)0,(0
20
0f
exp2
3
1
)0(det
)(detlnexp
2
3
0,
,
T
T
i
GC
C
iF
zRVezV
M
zMN
TZ
TZ
ieR
T
M
V
N
TR 2
2f detln
Saddle point: 0z
)0,(
)0,(0*
exp
exp),(ln1)(
T
TC
iF
iFzTZ
VT
saddle point reweighting factor
Similar to the reweighting method(sign problem & overlap problem)
Saddle point in complex /T plane
• Find a saddle point z0 numerically for each conf.
• Two problems– Sign problem– Overlap problem
0detln
0
f
zT
T
M
V
N
Technical problem 1: Sign problem
• Complex phase of detM– Taylor expansion (Bielefeld-Swansea, PRD66, 014507 (2002))
• || > /2: Sign problem happens. changes its sign.• Gaussian distribution
– Results for p4-improved staggered– Taylor expansion up to O(5)
– Dashed line: fit by a Gaussian function
Well approximated
0.23 T
histogram of 2
eW
)(detlnIm(phase) f MN
: NOT in the range [-, ]
ie
2Im 0
10tf
zzDNNVn
n
FFi eeee F22
Technical problem 2: Overlap problemRole of the weight factor exp(F+i)
• The weight factor has the same effect as when (T) increased.• */T approaches the free quark gas value in the high density
limit for all temperature.
3
2f3
1
TTN
T
free quark gas
free quark gas
free quark gas
Technical problem 2: Overlap problem
• Density of state method W(P): plaquette distribution
dPPWiF
dPPWiFz
TP
P
)(exp
)(exp)( 0*
)( 2exp)(exp 2 PWFPWiFPPP
linear for small Plinear for small P
)( exp eff PWP for small PSame effect when changes.
Reweighting for =6g-2
Change: 1(T) 2(T)
Distribution:
Potential:
PNSS gg 12site12 6)()(
12112 WeWW gg SS
+ =
212site1 ln6ln WPNW
site
2
eff
1
2
1
NdP
d
dP
FdPP
(Data: Nf=2 p4-staggared, m/m0.7, =0)
Effective (temperature) for 0
( increases) ( (T) increases)
'e det ,' )(f PPMDUPW gSN
Overlap problem, Multi- reweightingFerrenberg-Swendsen, PRL63,1195(1989)
• When the density increases, the position of the importance sampling changes.
• Combine all data by multi- reweighting
Problem:• Configurations do not cover all
region of P.• Calculate only when <P> is near
the peaks of the distributions.
Plaquette value by multi-beta reweighting peak position of the distribution ○ <P> at each
P
)0,(
)0,(
exp
exp
T
T
iF
iFPP
Chemical potential vs density
• Approximations: – Taylor expansion: ln det M– Gaussian distribution: – Saddle point approximation
• Two states at the same q/T– First order transition at T/
Tc < 0.83, q/T >2.3
• */T approaches the free quark gas value in the high density limit for all T.
• Solid line: multi-b reweighting• Dashed line: spline interpolation• Dot-dashed line: the free gas limit
Nf=2 p4-staggered, lattice4163
Number density
Summary• Effective potentials as functions of the plaquette value and the quark
number density are discussed.
• Approximation:– Taylor expansion of ln det M: up to O(6)– Distribution function of =Nf Im[ ln det M] : Gaussian type.– Saddle point approximation (1/V expansion)
• Simulations: 2-flavor p4-improved staggered quarks with m/m 0.7 on 163x4 lattice– Existence of the critical point: suggested.– High limit: /T approaches the free gas value for all T.– First order phase transition for T/Tc < 0.83, q/T >2.3.
• Studies near physical quark mass: important.– Location of the critical point: sensitive to quark mass