Toward the understanding of QCD phase structure at finite...
Transcript of Toward the understanding of QCD phase structure at finite...
Toward the understanding of QCD phase structure at finite temperature and density
Shinji Ejiri Niigata University
WHOT-QCD collaboration
S. Ejiri1, S. Aoki2, T. Hatsuda3, K. Kanaya2, Y. Nakagawa1, H. Ohno4, H. Saito2, and T. Umeda5
1Niigata Univ., 2Univ. of Tsukuba, 3RIKEN, 4Bielefeld Univ., 5Hiroshima Univ.
Osaka Univ., 2012/10/2
QCD phase structure at high temperature and density
Lattice QCD Simulations
• Phase transition lines • Equation of state
• Direct simulation: Impossible at µ≠0.
quark-gluon plasma phase
hadron phase
RH
IC
SPS
AG
S
color super conductor? nuclear
matter µq
T
RHIC E-scan J-PARC
LH
C
quarkyonic?
deconfinement?
chiral SB?
Histogram method Problem of Complex Determinant at µ≠0
Boltzmann weight: complex at µ≠0 Monte-Carlo method is not applicable.
Distribution function in Density of state method (Histogram method)
X: order parameters, total quark number, average plaquette etc.
Expectation values
( ) ( ) ,,, ,, µ=µ ∫ TmXWdXTmZ
( ) ( ) ( )( ) gSN emMXX-DUTmXW −µδ≡µ ∫ ,det ˆ ,,; f
[ ] ( ) [ ] ( ) ,,, 1,,
µ= ∫µTmXWXOdX
ZXO
Tm
( ) ( ) gSNTm
emMODUZ
O −
µµ= ∫ f),(det 1
,,complex
( ) ( )
( ) ( )( )
( )( ) ( )
( )'
0
'6
)0,(
)0,(0
'6f
0site
0
0
f
0site
0,det,det
ˆ
0,det,det ˆ
P
NPN
N
PN
mMmMe
P-P
mMmMP-P
ePR
µ≡
′δ
µ′δ
=′ β−β
=µβ
=µββ−β
(β, m, µ)-dependence of the Distribution function • Distributions of plaquette P (1x1 Wilson loop for the standard action)
( ) ( ) ( )( ) PNNmMP-PDUmPW ˆ6 sitef e ,det ˆ,,, βµ′δ≡µβ′ ∫
Effective potential:
( ) ( )[ ] ( ) ( )µββ−β=µβ−=µβ ,,,,ln0,,,,,ln,,, 0000effeff mmPRmPVmPWmPV
(Reweight factor) ( ) ( ) ( )0,,,,,,,,,, 0000 mPWmPWmmPR βµβ≡µββ
( ) ( ) ( )( )
P
N
mMmMPNPR
f
0,det,det ln6ln0
0site
µ+β−β=
independent of β0
PNSgˆ6 siteβ−=
• Sign problem: If changes its sign frequently,
Sign problem
( )( ) error) al(statistic
0,det,det~)(
fixed 0
<<µθ
X
i
mMmMeXW
θie
θ: complex phase of (detM)Nf
( )( )
( )( )
fixed 0
fixed 0
ff
0,det,det
0,det,det
X
Ni
X
N
mMmM e
mMmM µ
=
µ θ
Overlap problem
• W is computed from the histogram. • Distribution function around X where is minimized: important. • Veff must be computed in a wide range.
( ) ( ) ( )( ) lnexp1 1eff∫∫ +−== dXORXV
ZdXXORW
ZOR
)(ln)(eff XWXV −=
( )ORXV ln)(eff −
+ =
( )ORXV ln)(eff −( )ORln−)(eff XV
reliable range
Out of the reliable range
If X-dependence is large. reliable range
Distribution function in quenched simulations Effective potential in a wide range of P: required.
Plaquette histogram at K=1/mq=0. Derivative of Veff at β=5.69
dVeff/dP is adjusted to β=5.69, using
These data are combined by taking the average.
( ) ( ) ( )12site1eff
2eff 6 β−β−β=β N
dPdV
dPdV,4243
site ×=N 5 β points, quenched.
Distribution functions for ΩR and P
• If W(P,Ω) is a Gaussian distribution, – The peak position of W(P,Ω) (<P>,<Ω>) – The width of W(P,Ω) susceptibilities χP, χΩ
• If W(P,Ω) have two peaks, first order transition
4κ 4κ4
f* 48 KN+β=β
Expectation value of Polyakov loop and its susceptibility by the reweightuing method at µ=0.
4f
* 48 KN+β=β
lattice 4243 ×Ω
Ωχ
Ω
Ωχ~
Ω
µ-dependence of the effective potential
1st order phase transition
Critical point ( )µ,,eff TXV
Crossover
Correlation length: short V(X): Quadratic function
Correlation length: long Curvature: Zero
Two phases coexist Double well potential
)(ln)(eff XWXV −=( ) ( ),,, , µ=µ ∫ TXWdXTZX: order parameters, total quark number, average plaquette, quark determinant etc.
T
µ
hadron
QGP
CSC?
Quark mass dependence of the critical point
• Where is the physical point? • Extrapolation to finite density
– investigating the quark mass dependence near µ=0 • Critical point at finite density?
mud 0
0 ∞
∞
Physical point
1storder
2ndorder
Crossover
1storder
ms
Quenched Nf=2
0=µ
µ
Distribution function in the heavy quark region
• We study the properties of W(X) in the heavy quark region.
• Performing quenched simulations + Reweighting.
• We find the critical surface. • Standard Wilson quark action +
plaquette gauge action, • lattice size: • 5 simulation points; β=5.68-5.70. (WHOT-QCD, Phys.Rev.D84, 054502(2011))
( )( ) ( ) ( )( )( )+Ωµ+Ωµ⋅+=
µIR
Ns
N TiTKNPKNNM
KMN tt sinhcosh2122880,0det,detln 34
siteff
phase
4243 ×
Hopping parameter expansion
P: plaquette, Ω=ΩR+iΩI : Polyakov loop ( ) 10,0det =M
PNSg β−= site6
Effective potential near the quenched limit(µ=0) WHOT-QCD, Phys.Rev.D84, 054502(2011)
• detM: Hopping parameter expansion, • First order transition at K = 0 changes to crossover at K > 0.
Quenched Simulation (mq=∞, K=0)
K~1/mq for large mq
Critical point for Nf=2: Kcp=0.0658(3)(8)
02.0≈πm
Tc
( ) ( )[ ] ( ) ( )µββ−β=µβ−=µβ ,,,,ln0,,,,,ln,,, 0000effeff mmPRmPVmPWmPV
( ) ( ) ( )( )
P
N
mMmMPNPR
f
0,det,det ln6ln0
0site
µ+β−β=
at phase transition point
Order of the phase transition Polyakov loop distribution
54 100.2 :point Critical −×≈κ
Effective potential of |Ω| on the pseudo-critical line at µ=0
Ωχ
• The pseudo-critical line is determined by χΩ peak.
• Double-well at small K – First order transition
• Single-well at large K – Crossover
Polyakov loop distribution in the complex plane
• on βpc measured by the Polyakov loop susceptibility.
0.04 =κ 64 100.5 −×=κ 54 100.1 −×=κ
54 105.1 −×=κ 54 100.2 −×=κ 54 105.2 −×=κ
critical point
RΩ RΩ RΩ
RΩRΩ RΩ
IΩ IΩ IΩ
IΩ IΩ IΩ
(µ=0)
Distribution function of ΩR at finite density
• Hopping parameter expansion
• Adopting • Effective potential:
• V0 is Veff (µ=0) when we replace (at µ=0, )
( ) ( ) ( )( ) PNNRRR eMDUKW ˆ6 sitef det ˆ ,,, −κΩ−Ωδ=µβΩ ∫
( ) ( ) ( )0,;
3f0effeff
0lncosh2120,0,,,
=µ=βΩ
θ−Ωµ×−β=µβK
iR
Ns
N
R
tt eTKNNVKV
( )( )IN
sN TKNN tt Ωµ⋅=θ ˆsinh212 f
3
( ) ( )µβΩ−=µβΩ ,,;ln,,;eff KWKV RR
,48 4f0 KN+β≡β
( )0,;0
0ln ,,
=µ=βΩ
θ−µβ≡K
i
ReKV
( )TKK tt NN µ⇒ cosh
Phase-quenched part
( )( ) ( )( ) ( )[ ]
0,;
3fsite
4f0
0 0
ˆcosh212ˆ2886exp0,0,
,,=µ=βΩ
θ+Ωµ×−+β−β=β
µβKR
Ns
N
R
tt iTKNNPNKNW
KW
Phase average
( ) ( ) RN
sN tt KNNVKV Ω×−β=β 3
f0effeff 2120,0,0,,
• Sign problem: If changes its sign,
• Cumulant expansion
– Odd terms vanish from a symmetry under µ ↔ −µ (θ ↔ −θ) Source of the complex phase
If the cumulant expansion converges, No sign problem.
+θ+θ−θ−θ=
Ω
θ CCCCP
i iieR
432
, !41
!321exp
Avoiding the sign problem (SE, Phys.Rev.D77,014508(2008), WHOT-QCD, Phys.Rev.D82, 014508(2010))
error) al(statistic fixed ,
<<Ω
θ
RP
ie
=θθ+θθ−θ=θθ−θ=θθ=θΩΩΩΩΩΩΩ CPPPPCPPCPC RRRRRRR
43
,,,
2
,
332
,,
22,
,23 , ,
cumulants →0 →0
<..>P,ΩR: expectation values fixed P and ΩR.
θie
θ: complex phase MdetlnIm≡θ ( ) IN
sN TKNN tt Ωµ⋅≈ sinh212 f
3
Convergence in the large volume (V) limit The cumulant expansion is good in the following situations. • If the phase is given by
– No correlation between θx.
– Ratios of cumulants do not change in the large V limit. – Convergence property is independent of V, although the phase fluctuation becomes larger as V increases. – The application range of µ can be measured on a small lattice.
• When the distribution function of θ is perfectly Gaussian, the average of the phase is give by the second order,
∑θ=θx
x
θ= ∑Ω
θ
nC
nn
P
i
nie
R !exp
,
θ=≈
∑= ∑∑∏ Ω
θ
Ω
θ
Ω
θ
x nC
nx
n
xP
i
P
i
P
i
nieee
R
x
R
xx
R !exp
,,
,
)(~ VOx
C
nxC
n ∑ θ≈θ
θ−=
Ω
θ
CP
i
Re 2
, 21exp
Cumulant expansion
• The effect from higher order terms is small near the critical point of phase-quenched part.
69.5* =β +θ−θ+θ−=Ω
θ
CCC
i
Re 642
!61
!41
21 ln
( ) ( ) ( ) ( ) ( )TKTKK ttt NNN μsinhμμcoshμ0 cpcpcp >=
( ) ( ) 00002.0μsinhμ4 =TK
( ) ( ) 0001.0μsinhμ4 =TK
( ) ( ) 00005.0μsinhμ4 =TK
~0.00002
Effect from the complex phase factor • Polyakov loop effective potential for each
at the pseudo-critical (β, K). – Solid lines: complex phase omitted, i.e.,
– Dashed lines: complex phase is estimated with
( ) 0μsinh =T
( )TK tN µcosh
( ) ( )TT µ=µ coshsinh
The effect from the complex phase factor is very small except near ΩR=0.
( ) ( )
( )fixed:
20
fixed:0eff
21
ln
R
R
R
iRR
V
eVV
Ω
Ω
θ
θ+Ω≈
−Ω=Ω
( ) )sinh212( f3
IN
sN TKNN tt Ωµ⋅≈θ
Critical line in 2+1-flavor finite density QCD • The effect from the complex phase is very small for the determination of Kcp.
( ) ( ) ( )TKK tt NN µµ= cosh0 cpcp
Critical line for µu=µd=µ, µs=0 Critical line for µu=µd=µs=µ
The critical line is described by
( )( ) ( )+Ω×+=
R
Ns
N tt KNPKNM
KM 34site 2122882
0detdetln2
( )( ) ( )( )( )
( ) +Ω
+
×++=
R
sNs
udNuds
Nsud
sud
TK
TKNPKKN
MKMKM
tttμcoshμcosh22122288
0detdetdetln 344
site3
2
t2)fcp(2μcoshμcosh2 N
NsN
sudN
ud KT
KT
K tt==
+
Nf=2 at µ=0: Kcp=0.0658(3)(8)
Nf=2+1
(WHOT-QCD, Phys.Rev.D84, 054502(2011))
Distribution function in the light quark region WHOT-QCD Collaboration, in preparation,
(Nakagawa et al., arXiv:1111.2116)
• Perform phase quenched simulations • Add the effect of the complex phase by the reweighting. • Calculate the probability distribution function.
• Goal
– The critical point – The equation of state
Pressure, Energy density, Quark number density, Quark number susceptibility, Speed of sound, etc.
Probability distribution function by phase quenched simulation
• We perform phase quenched simulations with the weight:
( ) ( ) ( ) ( )( )
( ) ( ) ( )( )µβ×=
µ−δ−δ=
µ−δ−δ=µβ
θ
−θ
−
∫∫
,,,','
,det 'ˆ'ˆ
,det'ˆ'ˆ ,,,','
0','
f
f
mFPWe
emMeFFPPDU
emMFFPPDUmFPW
FP
i
SNi
SN
g
g
( ) gSN emM −µ ,det f
( ) ( )( )0det
detlnsite
f
MM
NNF µ
=µ
( ) ( ) ( ) PNN eMFFPPDUFPW ˆ60
sitef det'ˆ'ˆ ',' β−δ−δ= ∫
MN detlnImf≡θP: plaquette
Distribution function of the phase quenched.
expectation value with fixed P,F histogram
µ-dependence of the effective potential Curvature of the effective potential
+ =
1st order phase transition
( )[ ] ,ln 0 β− PW
Critical point
[ ] ln θ− ie( )[ ] ,ln 0 β− PW( )[ ] ,ln β− PW
+ =
Crossover
phase effect
phase effect
Curvature: Zero
Curvature: Negative
[ ] ln θ− ie
T
µ
hadron
QGP
CSC
Curvature of the effective potential • If the distribution is Gaussian,
( ) ( )
−
χ−
πχ×
−
χ−
πχ≈ 2sitesite2sitesite
0 2exp
226exp
26),( FFNNPPNNFPW
FFPP
( )2site6 PPNP −=χ
( ) ( )P
NFPP
Wχ
=∂−∂ site
20
2 6,ln ( ) ( )F
NFPF
Wχ
≈∂−∂ site
20
2
,ln
( )2site FFNF −=χ
at the peak of the distribution
Complex phase distribution • We should not define the complex phase in the range from -π to π. • When the distribution of q is perfectly Gaussian, the average of the
complex phase is give by the second order (variance),
• Gaussian distribution → The cumulant expansion is good. • We define the phase
– The range of θ is from -∞ to ∞.
( ) ( )( ) ( )
µ
µ∂
∂=
µ=µθ
µ
µ
∫ Td
TMN
MMN
T
0ffdetlnIm
0detdetlnIm
π −π π −π θ θ
W(θ) W(θ)
C
2θNo information
θ−=θ
CFP
ie 2
, 21exp
Distribution of the complex phase
• Well approximated by a Gaussian function. • Convergence of the cumulant expansion: good.
θ−≈θ 2
, 21exp
FP
ie
at the peak of W0 in each simulation
00 ,
2
,
2
21
21
µβθ≈θ
FP
Simulations
• Simulation point in the (β, µ0/T) • Peak of W0(P,F) for each µ
lattice 483 × 8.0≈ρπ mm
2-flavor QCD Iwasaki gauge + clover Wilson quark action Random noise method is used.
Curvature of the effective potential -lnW0
• The curvature for F decreases as µ increases.
Effect from the complex phase
• Rapidly changes around the pseudo-critical point.
Critical point at finite µ
• zero curvature: expected at a large µ.
Curvature of the effective potential • Without the complex phase effect
( ) ( )F
NFPF
Wχ
≈∂−∂ site
20
2
,ln ( )2site FFNF −=χ
Phase average • 2nd order cumulant
21ln
,
2
, FPFP
ie θ−≈θ00 ,
2
,
2
21
21
µβθ≈θ
FP
Curvature of the effective potential • The effect of the phase incruded.
zero curvature Critical point
Peak position of W(P,F) • The slopes are zero at
the peak of W(P,F).
( ) ( )
( ) ( ) ( )P
eFP
PRNFP
PW
P
eFP
PWFP
PW
FP
i
site
FP
i
∂
∂+µµ
∂∂
+β−β+µβ∂
∂=
∂
∂+µβ
∂∂
=µβ∂
∂
θ
θ
,0000
0
,0
ln,,,ln6,,,ln
ln,,,ln,,,ln
0ln ,0ln=
∂∂
=∂
∂FW
PW
( ) ( )( )
µβµβ
=µµ00
00 ,,,
,,,,,,FPWFPWFPR
( ) ( )
( ) ( )F
eFP
FRFP
FW
F
eFP
FWFP
FW
FP
i
FP
i
∂
∂+µµ
∂∂
+µβ∂
∂=
∂
∂+µβ
∂∂
=µβ∂
∂
θ
θ
,000
0
,0
ln,,,ln,,,ln
ln,,,ln,,,ln If these terms
are canceled,
( ) ( ) )const.(,,,,,, 000 ×µβ≈µβ FPWFPW
• W(β, µ) can be computed by simulations around (β0,µ0).
=0
=0
),(ln FPW−
QCD phase diagram
phase-quenched QCD finite-density QCD
pion condensed phase
mπ/2 µ
Τ
0=θiecolor super-conductor phase?
µ
Τ
( ) ( )µβ=×µβ θ ,,,,,,,,,0 mFPWemFPWFP
i
Summary • We studied the quark mass and chemical potential dependence
of the nature of QCD phase transition.
• The shape of the probability distribution function changes as a function of the quark mass and chemical potential.
• To avoid the sign problem, the method based on the cumulant expansion of θ is useful.
• Our results by phase quenched simulations suggest the existence of the critical point at high density.
• To find the critical point at finite density, further studies in light quark region are important applying this method.
Backup
Complex phase • Gaussian distribution → The cumulant expansion is good. • We define the phase
– The range of θ is from -∞ to ∞.
• At the same time, we calculate F as a function of µ,
• The reweighting factor is also computed,
( ) ( )( ) ( )
µ
µ∂
∂=
µ=µ
µ
µ
∫ Td
TMN
MMNF
T
0ffdetlnRe
0detdetln
( ) ( )( ) ( )
µ
µ∂
∂=
µ=µθ
µ
µ
∫ Td
TMN
MMN
T
0ffdetlnIm
0detdetlnIm
( ) ( )( ) ( )
µ
µ∂
∂=
µµ
=µµ
µ
µ∫ Td
TMN
MMNC
T
T0
detlnRedetdetln f
0f
Distribution function for P and ΩR
• Effective potential • Hopping parameter expansion
• 2 parameters in V0: – V0 is the same as Veff (µ=0) when
• 1 parameter in θ:
( ) ( ) ( ) ( )( ) PNNRRR eMPPDUPW ˆ6 sitef det ˆ ˆ ,,, −κΩ−Ωδ−δ=κβΩ ∫
( ) ( ) ( )( ) ( )R
tt
P
iR
Ns
N eTKNNPNKNVVΩ
θ−Ωµ×−+β−β−=β−κβ,
3fsite
4f00effeff lncosh21228860,,
( )( )IN
sN TKNN tt Ωµ⋅=θ ˆsinh212 f
3
( ) ( )κβΩ−=κβΩ ,;,ln,;,eff RR PWPV
,48 *4f β≡+β KN ( )TK tN µcosh
( )RP
ieVΩ
θ−κβ≡,0 ln ,
( )TKK tt NN µ⇒ cosh
( ) ( ) ( ) ( )TKTTKTK ttt NNN µ<µµ=µ coshtanhcoshsinh
Phase-quenched part
( )( )
( ) ( ) ( ) ( )( )
( ) ( )( ) ( )
( )RP
NPN
KRR
K
NPN
RR
MKMe
-PP-
MKMe-PP-
WKW
Ω
β−β
=µ=β
=µ=β
β−β
µ≡
ΩΩδδ
µΩΩδδ
=β
µβ
,
ˆ6
)0,(
)0,(
6
0
f
0site
0
0
f
0site
0,0det,det
ˆˆ
0,0det,det ˆˆ
0,0,,,
Order of phase transitions and Distribution function
• Peak position of W:
0effeff =Ω
=Rd
dVdP
dV
crossover 1 intersection
first order transition 3 intersections
Lines of zero derivatives for first order
( ) ( )κβΩ−=κβΩ ,;,ln,;,eff RR PWPV
( ) ( ) ( ) ( )( ) PNNRRR eMPPDUPW sitef 6 det ˆ ˆ ,,, −κΩ−Ωδ−δ=κβΩ ∫
Derivatives of Veff in terms of P and ΩR
• Contour lines of and at (β,κ) = (β0,0) correspond to
the lines of the zero derivatives at (β,κ).
( ) ( ) ( )( ) site4
f00effeff 28860,, NKN
dPdV
dPKdV
+β−β−=β
−β ( ) ( )
µ
×−=Ω
β−
Ωβ
TKNN
ddV
dKdV
tt Ns
N
RR
cosh2120,, 3f
0effeff
Phase-quenched part: when is neglected,
constant shift constant shift
dPdVeff
RddV
Ωeff
( ) ( ) ( )( ) ( ) RN
sN TKNNPNKNVV tt Ωµ×−+β−β−=β−κβ cosh21228860,, 3
fsite4
f00effeff
θieln
P PRΩRΩ
measured at κ = 0
dPdVeff
RddV
Ωeff
lines of and in the (P,Ω) plane
• Small K: lines of : S-shape first order • Large K: lines of : straight line crossover
0eff =dP
dV 0eff =ΩRd
dV
( ) ( ) ( )0*
site4
f0site0eff 64860,
β−β=+β−β=β NKNN
dPdV ( )
µ
×=Ω
βT
KNNd
dVtt N
sN
R
cosh2120, 3f
0eff
RΩRΩ
P P
0eff =dP
dV 0eff =ΩRd
dV
5.70 5.69 b*= 5.68
K4=0.0
K4=1.0E-5 K4=2.0E-5
K4=3.0E-5
0eff =ΩRd
dV
0eff =ΩRd
dV