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.