Phase structure of finite density Phase structure of finite density - - PowerPoint PPT Presentation

SLIDE 1

Phase structure of finite density Phase structure of finite density lattice QCD by a histogram method Q y g

Shinji Ejiri Shinji Ejiri

Niigata 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)

SLIDE 2

Phase structure of QCD at high temperature and density

Lattice QCD Simulations

quark-gluon plasma phase

• Phase transition lines
• Equation of state

RHI SP

T

LH

• Direct simulation:

C PS A RHIC low-E FAIR C

• Direct simulation:

Impossible at 0.

AGS deconfinement?

hadron phase

color super quarkyonic? chiral SB?

phase

color super conductor? nuclear matter

q

SLIDE 3

Probability distribution function

 Distribution function (Histogram)

X: order parameters, total quark number, average plaquette etc.

   

, , , , ,   



T m X W dX T m Z

histogram

 In the Matsubara formalism,



       

S N



     

g

S N

m M DU T m Z



  



e , det , ,

f

h d M k d i S i

       

g

S N

m M X X- DU T m X W



     



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).

SLIDE 4

-dependence of the effective potential

   



) ( ln ) (

eff

X W X V  

   ,

, , ,   



T X W dX T Z

X: order parameters, total quark number, average plaquette, quark determinant etc.

 

Crossover Critical point

 

 , ,

eff

T X V

Correlation length: short V(X): Quadratic function Correlation length: long

T

Correlation length: long Curvature: Zero

T

QGP

1st order phase transition

hadron CSC?

1 order phase transition

Two phases coexist D bl ll i l



CSC?

Double well potential

SLIDE 5

Quark mass dependence of the critical point



2ndorder 1storder

Quenched Nf=2



Physical point 2ndorder 1storder



y p Crossover

ms

1storder Crossover

 

mud

0 0



• Where is the physical point?
• Extrapolation to finite density

– investigating the quark mass dependence near =0

• Critical point at finite density?

SLIDE 6

Equation of State

• Integral method

– Interaction measure

ln 1 3 Z p     

e ac o easu e

computed by plaquette (1x1 Wilson loop) and the derivative of detM.

, ln

3 4

a VT T   P

– Pressure at =0

Z VT T p ln 1

3 4 

• Integral

 



       

a

a d T p T p T p ln 3

4 4 4

VT T

 





a a a

T T T

4 4 4

a0: start point p=0

3

) ( d t ) ( 1     M N Z

• Pressure at 0,

   

3 4 4

) ( det ) ( det ln ) ( ) ( ln 1

 

                      M M N N Z Z VT T p T p

s t

• with

) ( det ) ( det

• r

M M P X  

 

 

, , , 1

, ,

 





T m X W X dX Z X

T m

SLIDE 7

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 density

Application to the light quark region at finite density

– S.E., Phys.Rev.D77, 014508(2008)) – WHOT-QCD Collaboration, in preparation WHOT QCD Collaboration, in preparation (Lattice 2011 proc.: Y. Nakagawa et al., arXiv:1111.2116)

SLIDE 8

Distribution function in the heavy quark region

WHOT 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 quark space in the heavy quark region.

• Performing quenched
• Performing quenched

simulations + Reweighting. Reweighting.

– plaquette gauge action + Wilson quark action

SLIDE 9

(, m, )-dependence of the Distribution function

• Distributions of plaquette P (1x1 Wilson loop for the standard action)

       

P N N

m M P P- DU m P W

site f

6

e , det , , ,       



       

m M P P DU m P W e , det , , ,    



(Reweight factor

     

, , , , , , , , , , m P W m P W m m P R         

 

     

 

 

' 6 ) ( ' 6

f f

det , det , det

N P N N P N

m M m M m M P P-                

    

 

 

 

 

   

' ' 6 ) , ( ) , ( ' 6

site site

, det , det

P P N P N

m M m M e P P- e P R              

           

Effective potential:

   

 

   

            , , , , ln , , , , , ln , , ,

ff ff

m m P R m P V m P W m P V 

  

 

   

        , , , , ln , , , , , ln , , ,

eff eff

m m P R m P V m P W m P V

     

N

m M P N P R

f

, det l 6 l       

       

P

m M P N P R , det , ln 6 ln

site

           

SLIDE 10

Distribution function in quenched simulations

Effective potential in a wide range of P: required Effective 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

     

1 2 site 1 eff 2 eff

6        N dP dV dP dV , 4 243

site

  N

5  points, quenched. j  , g These data are combined by taking the average.

     

1 2 site 1 2

    dP dP

SLIDE 11

Effective potential near the quenched limit

WHOT QCD Phys Rev D84 054502(2011) WHOT-QCD, Phys.Rev.D84, 054502(2011) Quenched Simulation (mq=, K=0) first order

dP dV

eff

K~1/mq for large mq

dP

crossover Quark mass smaller

• detM: Hopping parameter expansion

lattice, 4 243 

5  points, Nf=2

• detM: Hopping parameter expansion,

Nf=2: Kcp=0.0658(3)(8)

02 T

   

 

             

R N s N

t t

K N P K N N M K M N

3 4 site f f

2 12 288 det det ln

l t f P l k l

• First order transition at K = 0 changes to crossover at K > 0.

02 . 



m Tc

  



real part of Polyakov loop

SLIDE 12

Endpoint of 1st order transition in 2+1 flavor QCD

 

 K M d

Nf=2: Kcp=0.0658(3)(8)

   

 

             

R N s N

t t

K N P K N M K M

3 4 site

2 12 288 2 det det ln 2

Nf=2+1

         

   

     

N N N s ud

M K M K M det det det ln

3 4 4 3 2

   

       

R N s N ud s N s ud site

t t t

K K N P K K N 2 2 12 2 288

3 4 4

The critical line is described by

t

2 2

N N N

K K K

t t  t

2) f cp(

2 2

N s ud

K K K

t t



 

SLIDE 13

Finite density QCD in the heavy quark region

       

x U x U x U x U

a a † 4 † 4 4 4

q q

e , e

  

 

in detM

 

   

   K M det

* *

q q

e ,      

   T T

e

Polyakov loop

   

   

     

 

                     

   N N T T N s N

T i T K N P K N N e e K N P K N N M K M N

t t t t

i h h 2 12 288 2 6 288 , det , det ln

3 4 * / / 3 4 site f f

     

 

         

I R N s N

T i T K N P K N N

t t

sinh cosh 2 12 288

3 4 site f

phase

i   

Polyakov loop

I R

i    

distribution

• We can extend this discussion

to finite density QCD. to finite density QCD.

SLIDE 14

Phase quenched simulations,

Isospin chemical potential

     

*

, det , det     K M K M

Isospin chemical potential

(Nf=2), u= d.

     

     , det , det , det

2

K M K M K M

          

                 

I R N s N

T i T K N P K N M K M

t t

sinh cosh 2 12 288 , det , det ln

3 4 site

phase

• If the complex phase is neglected,

  

 M , det

phase

Nf=2

 

T K K

t t

N N

  cosh

Nf=2 – Critical point:

     

N N  

   

cosh

cp cp

t t

N N

K T K   

           

t

N

T K K    cosh

cp cp

SLIDE 15

Distribution function for P and R at =0



         

g

S N R R R

e M P P DU P W



          



det ' ' , , ' , '

f

 

 

W  

     

 

 

   

N N P R N s N

t t R t t

K N N P N K N K N N P K N N P N W W                    

 3 4 fixed , 3 f 4 site f site

2 12 288 6 exp 2 12 288 6 exp , , 

 

   

R s K

N N P N K N        

f site f

2 12 288 6 exp

     

 

R N s N

t t

K N N P N K N V V             

3 f site 4 f eff eff

2 12 288 6 , ,

• Peak position of W:

eff eff

  

R

d dV dP dV

Lines of zero derivatives for first order

crossover 1 intersection first order transition 3 intersections

SLIDE 16

Derivatives of Veff in terms of P and R

l i 4 243

 i

In heavy quark region,

lattice, 4 243 

5  points

     

 

R N s N

t t

K N N P N K N V V             

3 f site 4 f eff eff

2 12 288 6 , ,

     

 

site 4 f eff eff

288 6 , , N K N dP dV dP K dV         

   

t t

N s N R R

K N N d dV d K dV

3 f eff eff

2 12 , ,        

hif

     

 

R s f site f eff eff

dP dP

R R

d d  

constant shift constant shift

• Contour lines of and at (,) = (0,0) correspond to

dP dVeff d dV 

eff

(, ) ( , ) p the lines of the zero derivatives at ().

dP

R

d

SLIDE 17

Lines of constant derivatives obtained by quenched simulations

At the critical point,

d /d dVeff/dP=0 dVeff/dR=0

R

R

Bl li dV /dP Blue lines: dVeff/dP Red lines: dVeff/dR

P

• The number of the intersection changes around =0.658

SLIDE 18

Finite density QCD in the heavy quark region

       

x U x U x U x U

a a † 4 † 4 4 4

q q

e , e

  

 

 

   

   K M det

* *

q q

e ,      

   T T

e

   

   

     

 

                     

   N N T T N s N

T i T K N P K N N e e K N P K N N M K M N

t t t t

i h h 2 12 288 2 6 288 , det , det ln

3 4 * / / 3 4 site f f

     

 

         

I R N s N

T i T K N P K N N

t t

sinh cosh 2 12 288

3 4 site f

phase

• We can extend this discussion to finite density QCD.

SLIDE 19

Finite density QCD

   

   

               

   T T N s N

e e K N P K N N M K M N

t t

2 6 288 det det ln

* / / 3 4 site f f

     

 

         

I R N s N

T i T K N P K N N

t t

sinh cosh 2 12 288

3 4 site f

phase

           

     

fixed , 3 f 4 site f site

sinh cosh 2 12 288 6 exp , ,

R t t

P I R N s N

T i T K N N P K N N P N W W



                

 

 

 

 

fixed , 3 f site 4 f

cosh 2 12 288 6 exp

R t t

P i R N s N

e T K N N P N K N

 

         

 

I N N

T K N N

t t

     sinh 2 12

3 f

     

 

 

fixed 3 f site 4 f eff eff

ln cosh 2 12 288 6 , ,

t t

P i R N s N

e T K N N P N K N V V

 

              

 

I s

T K N N    sinh 2 12

f

• Reweighting:

is a non-linear term in

 

fixed ,

R

P 

   .

, ,

ff ff

    V V

ln

i

e 

Reweighting: is a non linear term in

   .

, ,

eff eff

   V V

fixed ,

ln

R

P

e



Sign problem.

SLIDE 20

Avoiding the sign problem

(SE, Phys.Rev.D77,014508(2008), WHOT-QCD, Phys.Rev.D82, 014508(2010))

: complex phase

 

I N s N

T K N N M

t t

      sinh 2 12 det ln Im

3 f

• Sign problem: If changes its sign,

error) al (statistic 

 i

e

 i

e

• Cumulant expansion

error) al (statistic

fixed ,



R P

e

<..>F,P: expectation values fixed F and P.

p

              

 



C C C C P i

i i e

R

4 3 2 ,

! 4 1 ! 3 2 1 exp  

R

,

! 4 ! 3 2

          

4 3 2 3 3 2 2 2

2 3

cumulants

0 0

– Odd terms vanish from a symmetry under    

                  

       C P P F P C P P C P C

R R R R R R R

4 , , , 2 , 3 3 , , 2 2 ,

, 2 3 , ,

y y    

Source of the complex phase

If the cumulant expansion converges, No sign problem.

SLIDE 21

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=0 Distribution of  at K=0

        

 C F P i

e

2 ,

2 1 exp

B W( ) i h d b th f t

 

I N s N

T K N N

t t

     sinh 2 12

3 f

• Because W() is enhanced by the factor,

2



   

 

R N s N

T K N N W

t t

     cosh 2 12 exp ~ ,

3 f

the region of large R is important.

• The distribution of  seems to be of Gaussian at R>0.

c I



• The higher order cumulants etc. might be small.

c I 4



SLIDE 22

Convergence in the large volume (V) limit

• Because  ~ O(V), Naïve expectation:

– If so, the cumulant expansion does not converge.

? ) ( ~

n C n

V O 

However, this problem is solved in the following situation.

• The phase is given by



 

1



2



3



The 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



6



e eavy qua eg o , e p ase s e ag a y part of the Polyakov loop average, I.

4



5



6



8



7



9



8 7 9

correlation length

 

I N s N

T K N N

t t

     sinh 2 12

3 f

– 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 th the central limit theorem.

SLIDE 23

Convergence in the large volume (V) limit

This problem is solved in the following situation.

• The phase is given by



 

x

p g y

– No correlation between x.



x x

  

n i

i           

 

   x n C n x n x P F i P F i P F i

n i e e e

x x x

! exp

, , ,

       



 n C n n P F i

n i e ! exp

,

) ( ~ V O

x C n x C n

 

 

– Ratios of cumulants do not change in the large V limit. – Convergence property is independent of V g p p y p although the phase fluctuation becomes larger as V increases.

SLIDE 24

Effect from the complex phase

I 2



• The complex phase fluctuation is

l d d <0

c I

large around =0 and <0.

• However, the region around =0

is less important for finite 

I

d 2

is less important for finite .

dP d

R

P I 



, 2 R P I

d d

R

 

 ,

SLIDE 25

Critical surface at finite density

C t l t f th d i ti f V (P  )

 

T K

t

N

 i h

• Contour plot of the derivatives of Veff (P,R).

 

T K

t

N

   sinh

5

10 . 1



    

• Blue line is the dVeff/dR around the critical point
• Blue line is the dVeff/dR around the critical point.
• Effect from the complex phase is small on the critical line,

. 10 2

5 

  

   

 

N N dV K dV

Nt

    

 2 2 3 2

2 3

     

   

P N N N K N dP dV dP K dV

c I s Nt

             

 3 f 2 site 4 f eff eff

2 2 3 288 6 , ,

   

 

I N

N N dV K dV

t

        

 2 2 3 2

2 3

   

 

R c I s N s N R R

N N T K N N d dV d K dV

t t t

                    

f 3 f eff eff

2 2 3 cosh 2 12 , ,

SLIDE 26

Critical line in finite density QCD

(WHOT QCD Collab in preparation 11) (WHOT-QCD Collab., in preparation, 11)

• The effect from the complex

 

I N f s N

T K N iN

t t

     sinh 2 12

3

phase is very small for the determination of cp because

 

I f s



Critical line for u=d=

is small.

• On the critical line,

     

T K K

t t

N N

   cosh

cp cp

,

   

I N f s N

T K N N

t t

     tanh 2 12

cp 3

< 1

SLIDE 27

Distribution function in the light quark region

WHOT-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 function

We calculate the probability distribution function.

• Goal

– The critical point p – The equation of state

Pressure, Energy density, Quark number density, Quark number , gy y, Q y, Q susceptibility, Speed of sound, etc.

SLIDE 28

Probability distribution function by phase quenched simulation by phase quenched simulation

• We perform phase quenched simulations with the weight:

We perform phase quenched simulations with the weight:  

g

S N e

m M



 , det

f

               

       

 

 

, det ' ' , , , ' , '

f e

m M F F P P DU m F P W

S N i S N

g

       

         

  



' ' , det ' '

f

m F P W e e m M e F F P P DU

i S N i

g

 

    , , , ,

' , '

m F P W e

F P

 

det M 

expectation value with fixed P,F histogram

     

det det ln

f

M M N F   

M N det ln Im

f

 

P: plaquette      

P N N e

M F F P P DU F P W



     

site f

6

det ' ' ' , '

Distribution function

• f the phase quenched.

SLIDE 29

Phase quenched simulation

   

     



, , , , , , , ,

,

m F P W e m F P W

F P i

• When  = d pion condensation occurs

     

     , det , det , det

2

K M K M K M       ,

, det , det

*

    K M K M

• 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 QCD



 i

e

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,



 i

e

Near the phase boundary, – large fluctuations in : expected.

pion condensed phase

/2

 

ln   

  i i

e e

– W(P,F) and W0(P,F) are completely different. m/2



 

. ln

, ,

  

F P F P

e e

SLIDE 30

Peak position of W(P,F)

) , ( ln F P W 

• The slopes are zero at

the peak of W(P,F).

ln , ln       F W P W

   

P e F P P W F P P W

F P i

           

 ,

ln , , , ln , , , ln

     

              , , , , , , , , , F P W F P W F P R

     

P e F P P R N F P P W

F P i site

                

 ,

ln , , , ln 6 , , , ln

=0

   

F e F P F W F P F W

F P i

           

 ,

ln , , , ln , , , ln

If these terms are canceled

=0

   

F e F P F R F P F W F F F

F P i

               

 ,

ln , , , ln , , , ln

are canceled,

   

) const. ( , , , , , ,       F P W F P W

=0

• W() can be computed by simulations around (0,0).

SLIDE 31

Avoiding the sign problem

(SE, Phys.Rev.D77,014508(2008), WHOT-QCD, Phys.Rev.D82, 014508(2010))

: complex phase

M det ln Im  

• Sign problem: If changes its sign,

error) al (statistic 

 i

e

 i

e

• Cumulant expansion

error) al (statistic

fixed ,



F P

e

<..>P,F: expectation values fixed F and P.

p

              





C C C C F P i

i i e

4 3 2 ,

! 4 1 ! 3 2 1 exp  

,

! 4 ! 3 2

          

4 3 2 3 3 2 2 2

2 3

cumulants

0 0

– Odd terms vanish from a symmetry under    

                  

C F P F P F P F P C F P F P C F P C 4 3 , , , 2 , 3 3 2 , , 2 2 ,

, 2 3 , ,

y y    

Source of the complex phase

If the cumulant expansion converges, No sign problem.

SLIDE 32

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)

  1

phase is give by the second order (variance),

        

 C F P i

e

2 ,

2 1 exp

W W

C 2



No information

• Gaussian distribution  The cumulant expansion is good.

     

p g

• We define the phase

   

                   





d M N M N

T

det ln Im det ln Im

– The range of  is from - to .

     

                   





T d T N M N

f f

Im det ln Im

SLIDE 33

Distribution of the complex phase

4 . 7 . 1     T 4 5 . 1     T

3

 4 .   T

lattice 4 83 

8 . 

  m

m

2-flavor QCD Iwasaki gauge + clover Wilson

4 . 2 7 . 1     T 4 . 2 5 . 1     T

quark action

Random noise Random noise method is used.

• Well approximated by a Gaussian function.

Well approximated by a Gaussian function.

• Convergence of the cumulant expansion: good.

SLIDE 34

Distribution in a wide range

• Perform phase quenched

   

f

det

N

M F P W  

Reweighting method W0: distribution function in phase quenched simulations.

p q simulations at several points.

– Range of F is different.

Ch b i h i

         

fixed ,

det det , , , , ,

F P

M M F P W F P W F P R      

• Change  by reweighting

method.

• Combine the data
• Combine the data.

Distribution in a wide range: Distribution in a wide range:

• btained.
• The error of R is small

because F is fixed.

     

        , , ln , , , ln , , ln F P W F P R F P W

SLIDE 35

Summary

• We studied the quark mass and chemical potential dependence
• f the nature of QCD phase transition.

Q p

• The shape of the probability distribution function changes as a

p p y g function 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.