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.