View
221
Download
0
Tags:
Embed Size (px)
Citation preview
Finite Density Simulations:comparison of various approaches
orWarming-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.
μ
Critical end point
2SC CFL
T RHIC
GSI, JHFJ- PARC
· 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
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)
P. Braun-Munzinger, K. Redlich and J. Stachelin Quark Gluon Plasma 3 (nucl-th/0304013)
• A compilation of chemical freeze-out parameters appropriate for A-A collisions at different energies
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 (199
8) 53.– (gap energy) ~ μ
• Color-Flavor-Locking– M. Alford, K. Rajagopal and F. Wilczek Nucl.Phys. B537 (1999) 443.
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
Neutron Stars
1cm1000x1000
x1000 x
Finite density even in normal Nuclear
Matter ?
ρ~ fm^3
ρ
T
ρ
Using lattice QCD, we want to study
here !
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
Map of Wonder World of High Density
Sign Problem
Two-Color
I
<Tri-Critical Point
CSC
Yes, I will study this wonderful world by lattice
QCD !
2SCQCD as a function of T and μ・ Interesting
and sound physics from theoretical and experimental point of views.
Lattice QCD should provide fundamental information as a first principle calculation.
Lattice QCD with Chemical Potential
A natural way to introduce the chemical potential
iPP 44
( )( ) iaA xU x e
x ˆx ip
ˆ ,ip a
x x e
ˆ ˆ4 44, 4,( ) ( ) a
x x x xU x U x e
GSedet1 DUZ
)()( eTre GSNH DDUD
0 mD
550 mD complex :det
0 † detdetdet)(det 55
* real :det
),()( xUexU tt
)()( xUexU tt††
0
At
At
( )( ) iaA xU x e
ZGS /edet
In Monte Carlo simulation, configurations are generated according to the Probability:
Monte Carlo Simulationsvery difficult !
det : !Complex
1 det GSO DU O e
Z
Several Cases where we do not suffer from the Complex Determinant
1. Imaginary Chemical Potential
†0( ) D m
5 0 5 5 5( )D m
0( ) D m
0( )I Ii D m i †
0 5 5( ) ( )I I Ii D m i i
Roberge and Weiss, Nucl. Phys. B275[FS17](1986)734-745
( , ) ( , )iti ix t e x t
( , ) ( , )it
i ix t e x t Change variables as
1 1t
At the temporal Edge
Imaginary Chemical Potential
It can be considered as a special boundary condition.
2 2t
tN tt N
The Gauge action has Z3 invariance2
( ) ( )3
Z Z
All information is contained in2
03
4( , ) ( , ) ( , 1)i i ix t e U x t x t
4( , ) ( , ) ( , 1)i i ix t U x t x t
4 ( , )tNi te U x t N
/
4 ( , )Ti te U x t N
4 ( , )ii te U x t N
• Two-flavors (u and d) have opposite sign of the chemical potential:
2. Finite-Isospin (Iso-vector Chemical Potential)
If u d det ( )det ( ) det ( )det ( )u d u u
2†5 5det ( )det ( ) det ( )u u u
In other word
Phase Quench QCD Finite Iso-spin Model
3. Two-Color Model
• For Color SU(2) case,
),(det)),((det *** UU
22* UU )2(SUU
),(det),(det 2*
2 UU
:)(det Real !
for
{ , } 2 { , } 2
4. Quench Simulation ?
• Barbour et al. found
• Stephanov shows
Quench
0
Barbour et al., Nucl. Phys. B275 (86) 296
2c
m
(not )3N
c
m 0c
in the Chiral limit
Quench QCD
fN 0 limit of QCD
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 ?
G
G
G
G
Si
S
S
Si
eDU
DU
DU
eODU
edet
edet
edet
edet.
O
0
0
det
det
i
i
e
Oe
if the phase fluctuate rapidly.
Difficult to study the real QCD because of the Sign Problem !
Multi-parameter Reweighting)(e)(det
1 gSDUOZ
O
)0(det
)(dete)0(dete
1 )()()( 00 ggg SSSDUOZ
The Pessimism was wiped off by Fodor-Katz ! (2002)
Numerical Challenge: How to calculate det ( )Determinant of Giant Sparse Matrix
Gibbs, Phys.Lett. B172 (1986) 53
Large Sparse Matrix Smaller Dense Matrix6
1
det ( ) ( )s
t s t
VN V N
ii
e e
i : eigen values of a 6 6s sV V matrix which does not depends on
Fodor and KatzMulti-parameter reweighting technique
Allton et al. (Bielefeld-Swansea)
Taylor expansion at high T and low
n
n
n
n
n
)0(detln
!)0(det
)(detln
1
Fodor-Katz, JHEP03(2002)014
TE E 160 35 725 35. , MeV MeV
12FN2.0 ,025.0, sdu mm
8 ,6 ,4 ,43 ss NN
Standard gauge + Staggered fermion
162 2 ,ET MeV 360 40E MeV
Allton et al. (Bielefeld-Swansea)
Improved action + Improved staggered fermion
4163
MeV
a=0.29
0.2 ,1.0qm
Imaginary Chemical Potential deForcrand and Philipsen hep-lat/0205016
Im
)()( 210
I
IIC acca
(D’Elia and Lombardo hep-lat/0205022) At small
)()(log 644
220 OaaaZ )()(log
644
220 IIII OaaaZ ImI
complex:det real:detM
ReIm i
Standard gauge + Staggered fermion
,2FN 250.0qm46 ,48 33
3
I Z(3) symmetry
Color SU(2) QCD
• No Complex Determinant Problem here !
• Poor person’s QCD– Asymptotic free Non-Abelian Gauge
theory– Confinement/Deconfinement transiti
on• ’t Hooft’s monopole picture: SU(2) part i
s essential.
• But Baryons are qq states, not qqq !
SU(3)
SU(2)
Analyses of Two-color QCD• SU(2) lattice gauge theory at 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 Skulleru
d , 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 (hep-lat/0105026)
Kogut, Toublan, and Sinclair hep-lat/0205019
Muroya, Nakamura, Nonaka (hep-lat/001007, hep-lat/0111032, hep-lat/020
8006, Phys. Lett. B551 (2003) 305-310 )• Wilson fermion, Link-by-Link update
0
Standard gauge + Staggered fermion
05.0 ,612 ,48 ,8 334 m05.0 ,612 ,16 34 m
Evidence of di-quarkcondensation
Vector meson at Finite
Periodic boundary condition
Vector meson mass becomes small !
(This reminds us of CERES experiment.)
Muroya, AN, Nonaka
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
2
6exp( )GZ DUD D S
g
Then we can integrate over U.
Bilic et al. Nucl. Phys. B377 (92)615
Many useful formulae in Rossi and Wolff, Nucl.Phys. B248 (1984) 541
Strong Coupling Calculation (cont)
• Recent progress:– Nishida, Fukushima and Hatsuda, Phys.Rept. 398 (2004) 281 (S
U(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.)
( ) ( ) GSGE DU E S e ρ
1( )
EO dE E O
Zρρ
1( ) ( )
( )GS
GEO DU E S O U e
E
ρ where
( )Eρ
E
Histogram Smoothing
Density of States Method (2)
• Gocksch proposed to use the phase of the determinant instead of
GS
( ) ( ( )) det GSE DU E U e ρ 1
( ) iE
EO dE E e O
Zρρ
( ) iEZ dE E eρ ρ ( )Eρ
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. D6
6 (02) 106008,– Ambjorn, Anagnostopoulos, Nishimura and Verb
aarschot, J. HEP,10 (02) 062.
Complex Langevin
• Parisi, Phys. Lett. B131 (83) 383• Karsch and Wyld, Phys. Rev. Lett. 21 (85) 2242
dA S
d A
: Langevin Time : Gaussina White Noise
No Probability appears (explicitly)Only Eq. of MotionBut 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
/( / ) ( )T NN
N
Z T e Z
2
0
1( / )
2NZ d Z i T
Random Matrix Theory
• No dynamics, but a good theoretical framework.• Many activities: Akemann, Osborn, Splittorf, Tou
blan, Verbaarschot
†1
2†lim det
TrWW
N
m iWZ dW e
iW m
Density Profiles of Dirac operator eigenvaluesAkemann 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/0411030Eigenvalue-Distribution for Unquech SU(3) by Random Matrix
5mV 2.5F V 2fN
Random-Matrix Model Calculation by Klein et al.
SU(3)
SU(2)
2 21 1G uu G m
2 22 2G dd G m
2 25 5 / 2G u d d u Gρ
Taylor Expansion of Screening Masses (QCD-TARO)
22
20 0
1( ) (0)
2
dM d MM M
d d
2
2
0
d MT
d
2
2
0
d MT
d
Pseudo-Scalar Meson 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 ( )det ( ) det ( )det ( )u d u u 2†
5 5det ( )det ( ) det ( )u u u
2 2 2det ( ) det ( ) ie QCD
Finite Isospin (Iso-Vector) Model
· Difficulty of large Chemical Potentialand Great Trick by Fodor-Katz
Towards large density QCD; What we should do
mRe
Im
0
Eigen Value Distribution
( 0) D m : anti-HermiteD
When increases
Eigen Value Distribution
Re
Im
0 m
μConjugate Gradient to calculate
does not converge
max
min
1( )
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)
( )FT TjS j 1
2( )1
2
TT
T
j
j
2
det GSTZ DU ej
All full QCD update algorithms require
Fodor-Katz algorithm does not calculate , but evaluate
1( )
1( )
det ( )
det (0)
Concluding Remarks• Great Progress in Lattice QCD at finite Density in these years.
Talks today and tomorrow
• Most Important Point:– 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.T
RHIC
CFL
GSI, JHFJ- PARC
μ2SC