Transcript Finite Density Simulations: comparison of various

Finite Density Simulations:
comparison of various approaches
or
Warming-Up for Talks by
Fodor, deForcrand, Ejiri, Gavai, Lombardo,
Schimidt and Splittorff
Quantum Fields
in the Era of Teraflop-Computing
Nov. 22-25, 2004 ZiF, Bielefeld
Atsushi Nakamura, RIISE, Hiroshima Univ.
Plan of the Talk
• Introduction
– Motivation
– Formulation
• Complex Fermion Determinant
• Lattice Approaches today
• Old and New Ideas
• Discussions for the Next Step
– Phase controls Phase ?
Introduction
QCD as a function of T and μ.
· Now that we possess a theory of the strong
interactions, it is natural to explore the
properties of hadronic matter in unusual
environments, in particular at high
temperature or high baryon density.
•
There are three places where one might look
for the effects of high temperature and/or
large baryon density
T RHIC
Critical end point
1. the interior of neutron stars
2. during the collision of heavy ions at very
high energy per nucleon
3. about 10^-5 sec after the big bang
Gross, Pisarski and Yaffe, Rev.Mod.Phys. 53 (1981)
GSI,
J- PARC
JHF
2SC CFL
μ
P. Braun-Munzinger, K. Redlich and J. Stachel
in Quark Gluon Plasma 3 (nucl-th/0304013)
• A compilation of chemical freeze-out parameters appropriate
for A-A collisions at different energies
A Comparison with Lattice Results
P. Braun-Munzinger, K. Redlich and J. Stachel
T [MeV]
mB [GeV]
Color Super Conductivity
• Original Color Super Conductivity
–
–
–
–
B.C. Barrois Nucl.Phys.B129 (1977) 390
D. Bailin and A. Love, Phys.Rep. 107 (1984) 325.
M. Iwasaki and T. Iwado, Phys.Lett. B350 (1995) 163
(gap energy)～μ/1000
• Revival
– M. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B422(1998) 247.
– R. Rapp, T. Schaefer, E. V. Shuryak, M. Velkovsky,Phys. Lett. 81 (1998)
53.
– (gap energy)～μ
• Color-Flavor-Locking
– M. Alford, K. Rajagopal and F. Wilczek Nucl.Phys. B537 (1999) 443.
Neutron Stars
Super-Nova Explosion at the last stage
of the Evolution of Stars
4M < M < 8M
Neutron Star
Central Region
1cm x 1cm x 1cm
~ 109 ton
1000x1000
1cm
x1000 x
Finite density even in normal Nuclear Matter ?
T
Using lattice QCD,
we want to study
here !
0
ρ0=0.16～0.17/fm^3

Compressed Baryonic Matter Workshop, May 13-16, 2002, GSI Darmstadt:
H. Appelshaeuser, Dileptons from Pb-Au Collisions at 40 AGeV
http://www.gsi.de/cbm2002/transparencies/happelshaeuser1/index.html
Larger enhancement at 40 AGeV compared to 158 AGeV
KEK-PS E325 Collaboration
taken from Prof.Akaishi’s talk
Map of Wonder World of High Density
CSC
Sign
Problem
Two- <yy>
Color
Tri-Critical
Point
m
I
2SC
Yes, I will study
this wonderful
world by lattice
QCD !
QCD as a
function of T
and μ
・ Interesting
Lattice QCD should provide
fundamental information as a
first principle calculation.
and sound
physics from
theoretical
and
experimental
point of views.
Lattice QCD with Chemical Potential
U m ( x) = e
 m  ipm
x
iaAm ( x )
 x mˆ , x  e
x  mˆ
ipm a
A natural way to introduce
the chemical potential
P4  P4  im
ma
U 4 ( x) x4,ˆ x  U 4 ( x)e  x4,ˆ x
 = Tre
  ( H  mN )
U m ( x) = e
=  DUD y Dy e
1
=  DU det  e SG
Z
iaAm ( x )
 ( S G y y )
U t ( x)  e mU t ( x),
Ut ( x)  e mUt ( x)
†
 = D    m  m 0
At m = 0
†
det  : real
(det )* = det † = det  5 5 = det 　At m  0
 = D    m  m 0   5 5
det  : complex
1
 SG
O =  DU O det e
Z
In Monte Carlo simulation, configurations
are generated according to the
Probability: det  e  SG / Z
det  : Complex!
Monte Carlo Simulations
very difficult !
Several Cases where we do not suffer
from the Complex Determinant
( m ) = D    m  m 0
 ( m ) =  D    m  m 0
=  5  D    m  m 0   5 =  5 ( m ) 5
†
1. Imaginary Chemical Potential
(im I ) = D   m  im I  0
 (im I ) =  D    m  im I  0 =  5 (im I ) 5
†
Roberge and Weiss, Nucl. Phys. B275[FS17](1986)734-745
Change variables as
y ( x , ti )  e  t my ( x , ti ) y ( x , ti )  e  t my ( x , ti )
i
i
m
y ( x , ti )e U 4 ( x , ti )y ( x , ti  1)
 y ( x , ti )U 4 ( x , ti )y ( x, ti  1)
At the temporal Edge
Nt m
e U 4 ( x , ti = N t )
m /T
= e U 4 ( x , ti = N t )
Imaginary Chemical Potential
i
e U 4 ( x , ti = N t )
It can be considered as a special
boundary condition.
t Nt = N t
t2 = 2
t1 = 1
The Gauge action has Z3 invariance
2
Z ( ) = Z ( 
)
3
2
All information is contained in 0
2. Finite-Isospin (Iso-vector Chemical Potential)
• Two-flavors (u and d) have opposite sign
of the chemical potential:
If mu =  md
det ( mu )det ( md ) = det ( mu )det ( mu )
= det ( mu )det  5( mu )  5 = det ( mu )
†
In other word
Phase Quench QCD  Finite Iso-spin Model
2
3. Two-Color Model
• For Color SU(2) case,
U m =  2U m 2 for U m  SU(2)
*
(det (U ,  m )) = det (U ,  m ) *
*
*
= det  2 (U ,  m ) 2 = det (U ,  m ) *
{ m ,  } = 2 m
{ m  ,   } = 2 m
det (m ) : Real !
4. Quench Simulation ?
• Barbour et al. found
m
mc =
2 mN
(not mc =
)
3
mc  0
Quench
 =0
in the Chiral limit
• Stephanov shows
Quench QCD 
Nf  0 limit of QCD
Barbour et al., Nucl. Phys.
B275 (86) 296
Plan of the Talk
• Introduction
• Lattice Approaches today
– Reweighting
– Taylor Expansion
– Imaginary Chemical Potential
– Two-Color
• Old and New Ideas
• Discussions for the Next Step
– Phase controls Phase ?
Difficult to study
the real QCD
because of the
Sign Problem !
 O >=
 SG
i
DU
.
O
det

e
e

 DU det  e
=
 SG
 Oei > det
 ei > det

 SG
DU
det

e

 SG
i
DU
det

e
e

0

0
if the phase  fluctuate rapidly.
The Pessimism was wiped off by Fodor-Katz !
(2002)
Multi-parameter Reweighting
1
S g (  )
O =  DUO det ( m ) e
Z
1
S g ( 0 )
S g (  0 )  S g (  ) det ( m )
=  DUO e
det (0) e
Z
det (0)
Numerical Challenge: How to calculate
det ( m )
Determinant of Giant Sparse Matrix
Gibbs, Phys.Lett. B172 (1986) 53
Large Sparse Matrix
Smaller Dense Matrix
det ( m ) = e
i
NtVs m
6Vs
 (  e
i =1
: eigen values of a
6Vs  6Vs
i
 Nt m
)
matrix which does not depends on m
Fodor and Katz
Multi-parameter reweighting
technique
Allton et al. (Bielefeld-Swansea)
Taylor expansion at high T
and low m
 det (m )   m n  n ln det (0)
 = 
ln
n
det

(
0
)
n
!

m

 n=1
Fodor-Katz,
JHEP03(2002)014
TE = 160  35
. MeV, mE = 725  35 MeV
TE = 162  2 MeV,
m E = 360  40 MeV
Standard gauge
+ Staggered fermion
NF = 2 1
mu,d = 0.025, ms = 0.2
Ns3  4, Ns = 4, 6, 8
Allton et al. (Bielefeld-Swansea)
170 MeV
Improved action
+ Improved staggered fermion
163  4
m
ma=0.29
mq = 0.1, 0.2
Imaginary Chemical Potential
deForcrand and Philipsen hep-lat/0205016
(D’Elia and Lombardo hep-lat/0205022)
At small m
log Z (m ) = a0  a2 m 2  a4 m 4  O(m 6 ) m I = Im m
det  : complex
mI 

3
log Z ( m I ) = a0  a2 m I  a4 m I  O( m I )
det M : real
Z(3) symmetry
Im m = i Re m
 C (am I ) = c0  c1 (am I ) 2
m I = Im m
Standard gauge
+ Staggered fermion
N F = 2, mq = 0.025
83  4, 63  4
2
4
6
m=0.0
m=0.2
deForcrand-Philipsen
For small m, we may
have a look of the phase
transition line.
Color SU(2) QCD
• No Complex Determinant
Problem here !
• Poor person’s QCD
– Asymptotic free Non-Abelian
Gauge theory
– Confinement/Deconfinement
transition
SU(3)
• ’t Hooft’s monopole picture: SU(2) part
is essential.
• But Baryons are qq states, not
qqq !
SU(2)
Analyses of Two-color QCD
• SU(2) lattice gauge theory at m  0
 Nakamura (PLB140(1984)391)
• The first calculation, Pseudo-Fermion Method
 Hands,Kogut,Lombardo and Morrison (NPB558(’99)327)
• Staggered fermion, HMC and Molecular dynamics
 Hands,Montvay,Morrison,Oevers,Scorzato and
Skullerud , Eur.Phys.J. C17 (2000) 285 (hep-lat/0006018)
• Staggered fermion, HMC and Two-Step Multi-Boson algorithm
 Kogut, Toublan and Sinclair PLB514 (2001) 77 (hep-lat/0104010)
 Kogut, Sinclair, Hands and Morrison ,PRD64(2001)094505 (heplat/0105026)
 Kogut, Toublan, and Sinclair hep-lat/0205019
 Muroya, Nakamura, Nonaka (hep-lat/001007, hep-lat/0111032, heplat/0208006, Phys. Lett. B551 (2003) 305-310 )
• Wilson fermion, Link-by-Link update
Evidence of di-quark
condensation
Standard gauge
+ Staggered fermion
84 , 83  4, 123  6, m = 0.05
164 , 123  6, m = 0.05
Vector meson at Finite m
Periodic boundary condition
Muroya, AN,
Nonaka
Vector meson mass
becomes small !
(This reminds us of
CERES experiment.)
Plan of the Talk
• Introduction
• Lattice Approaches today
• Old and New Ideas
–
–
–
–
–
–
–
Strong-Coupling Expansion
Density of State
Complex Langevin
Canonical Ensemble
Random Matrix
Finite Iso-spin
Meron-Cluster
• Discussions for the Next Step
– Phase controls Phase ?
Strong Coupling Calculation
6
Z =  DUDy Dy exp(yy  2 SG )
g
Then we can integrate over U.
Many useful formulae in
Rossi and Wolff, Nucl.Phys. B248 (1984) 541
Bilic et al. Nucl. Phys. B377 (92)615
Strong Coupling Calculation (cont)
• Recent progress:
– Nishida, Fukushima and Hatsuda, Phys.Rept. 398 (2004) 281
(SU(2))
– Nishida, PRD69(04)094501(SU(3)), hep-ph/0310160(SU(2))
• KS-fermions, including the di-quark condensation
• Finite-Isospin is also considered (8-flavors)
Di-quark
Condensate
Chiral
Condensate
SU(2)
Density of States Method
• Original Form (We consider the quench case.)
 ( E ) =  DU  ( E  SG )e
1
O =
dE ( E ) O

Z
  SG
where
 (E)
O
E
E
1
  SG

DU

(
E

S
)
O
(
U
)
e
G
 (E) 
Histogram
E
Smoothing
Density of States Method (2)
• Gocksch proposed to use the phase of the
determinant instead of SG
 ( E ) =  DU  ( E   (U )) det  e
1
iE
O =
dE ( E )e O E

Z
iE
Z  =  dE ( E )e
 (E)
  SG
E =
Density of States Method (3)
• Most(?) general form was given
– in Muroya et al., Prog.Theor.Phys.qq0 (03) 615.
Sect. 5.5
• Recently a sophisticated version is proposed,
– Anagnostopoulos and Nishimura, Phys. Rev.
D66 (02) 106008,
– Ambjorn, Anagnostopoulos, Nishimura and
Verbaarschot, J. HEP,10 (02) 062.
Complex Langevin
• Parisi, Phys. Lett. B131 (83) 383
• Karsch and Wyld, Phys. Rev. Lett. 21 (85) 2242
S
=

d
Am
 : Langevin Time
 : Gaussina White Noise
dAm
No Probability appears (explicitly)
Only Eq. of Motion
But it converges sometimes in a wrong way
Canonical Ensemble instead of
Grand Canonical Ensemble
• Miller and Redlich, Phys.Rev.D35(87)2524
• Engels et al., Nucl. Phys. B558 (99) 307.
• K-F. Liu, hep-lat/0312027

Z (m / T ) =
 (e
N =
1
ZN =
2

2
0
m /T N
) ZN
d Z (i  m / T )
Random Matrix Theory
1

iW

m

 2TrWW †
Z = lim  dW det  †
e

N 
m 
 iW  m
m
• No dynamics, but a good theoretical framework.
• Many activities: Akemann, Osborn, Splittorf,
Toublan, Verbaarschot
Density Profiles of Dirac operator eigenvalues
Akemann et al. hep-lat/0409045
SU(2), Quench. For SU(3) See Akemann and Wettig, Phys.Rev.Lett. 92
(2004) 102002
Akemann, Osborn, Splittorff, Verbaarschot, hep-th/0411030
Eigenvalue-Distribution for Unquech SU(3) by Random Matrix
Nf = 2
mV  = 5 m F V = 2.5
Finite Isospin
• Son-Stephanov
m
m
mI
m
Kogut-Sinclair
Random-Matrix Model Calculation by Klein et al.
SU(2)
SU(3)
G 2 1 = uu  G 2m1
G 2 2 = dd  G 2 m2


G 2  = u  5d  d  5u / 2  G 2
Taylor Expansion of Screening Masses
(QCD-TARO)
2
dM
1 2d M
M ( m ) = M (0)  m
 m
 
2
dm 0 2
dm 0
d 2M
T
dm2
0
Pseudo-Scalar Meson
d 2M
T
dm2
0
Vector Meson
Meron-Cluster Algorithm
• Swendsen-Wang’s collective Monte Carlo
method.
• (+)+(-)=0 flips can be identified, and
• It works excellently for Spin system
• No one knows how to extend to the gauge
system.
Plan of the Talk
•
•
•
•
Introduction
Lattice Approaches today
Old and New Ideas
Discussions for the Next Step
– Phase controls Phase ?
Finite Isospin Model vs. QCD
Finite Isospin model = Two-flavor QCD with
Phase Quenching
 det (m ) 
QCD
2
= det ( m ) e
2
2 i
Finite Isospin (Iso-Vector) Model
det ( mu )det ( md ) = det ( mu )det ( mu )
= det ( mu )det  5( mu )  5 = det ( mu )
†
2
The difference is due to the Phase !
•
Sinclair, Talk at FDQCD at Nara, hep-lat/0311019
e
i / 2
Phase Contour
Ejiri, Phys.Rev. D69 (2004) 094506 (hep-lat/0401012)
Results with and without Phase
•
Nakamura, Takaishi and Sasai, to be publishded
Toussaint
1990
Towards large density QCD; What we should
do
· Difficulty of large Chemical Potential
and Great Trick by Fodor-Katz
( m = 0) = D    m
D  : anti-Hermite
Im
Eigen Value Distribution
Re
0
m
When m increases
Im
μ
0
 max

 min
Re
m
Conjugate Gradient to
calculate
( m )
1
does not converge
Eigen Value Distribution
This should occur also in SU(2) case.
It seems that it does not occur if one
introduce the di-quark source terms
(Kogut-Sinclair)
S F = y ( m )y  j y y  jyy

T

j

= (y y T ) 
  1 T
 2
1 
 T
2 y 
 
  y 
j

Z =  DU det   j e
T
2  SG
T
All full QCD update algorithms
require ( m )1
Fodor-Katz algorithm does not
1
calculate ( m ) , but evaluate
det  ( m )
det  (0)
Concluding Remarks
• Great Progress in Lattice QCD at finite Density in these years.
Talks today and tomorrow
• Most Important Point:
•
•
•
•
T
–
–
–
–
We have regions where we study the finite density world by lattice QCD,
i.e., finite density and finite temperature
We are lucky ! This is Region which RHIC is exploring.
We would like to go larger density region which GSI will study.
Technical progress is large
Model calculations have improved our understanding a lot
Finite Density Lattice QCD is still in Stone-Age.
We should work much more to understand
what is really problem.
RHIC
GSI,
J- PARC
JHF
2SC CFL
μ