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

A Comparison with Lattice ResultsP. Braun-Munzinger, K. Redlich and J. Stachel

T [MeV]

[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 (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

KEK-PS E325 Collaboration (taken from Yakkaichi’s presentation at Nuclear Chiral Dynamics, KEK 04)

taken from Prof.Akaishi’s talk

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

deForcrand-Philipsen

For small , 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 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

Finite Isospin

• Son-Stephanov

m

m

I

m

Kogut-Sinclair

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

The difference is due to the Phase !

• Sinclair, Talk at FDQCD at Nara, hep-lat/0311019

/ 2ie

Phase Contour Ejiri, Phys.Rev. D69 (2004) 094506 (hep-lat/0401012)

• Nakamura, Takaishi and Sasai, to be publishded

Results with and without Phase

Toussaint 1990

· 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