NF = 3 Critical Point from Canonical Ensemble

χQCD Collaboration: A. Li, A. Alexandru, KFL, and X.F. Meng

• Finite Density Algorithm with Canonical Approach and Winding Number Expansion

• Update on NF = 4, and 3 with Wilson-Clover Fermion

• New Algorithm Aiming at Large Volumes

A Conjectured Phase Diagram

Canonical partition function

)(det),,( 2][ UMeDUTkVZ kUS

CG

),,(21),,(

get we),,(),,( expansion fugacity theUsing

2

0

4

4

TTiVZedTkVZ

eTkVZTVZ

GCik

C

Vk

Vk

kT

CGC

4

Standard HMC Accept/Reject Phase

Canonical approach

Continues Fourier transformUseful for large k

Canonical ensembles

Fourier transform

K. F. Liu, QCD and Numerical Analysis Vol. III (Springer,New York, 2005),p. 101.Andrei Alexandru, Manfried Faber, Ivan Horva´th,Keh-Fei Liu, PRD 72, 114513 (2005)

WNEM

Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

2 21det ( ) det ( )2

ikk M U d e M U

Real due to C or T, or

CH

Winding number expansion in canonical approach to finite density Xiangfei Meng - Lattice 2008 Williamsburg

Winding number expansion (I)In QCD Tr log loop loop expansion

In particle number space

Where

T

S

T

S

T

S

T

S

A0 A0

A1 A2

8

Observables

Polyakov loop

Baryon chemical potential

Phase

Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

9

Phase diagram

T

ρ

coexistenthadrons

plasma

Four flavors

T

ρ

coexistenthadrons

plasma

Three flavors

?

Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

Baryon Chemical Potential for Nf = 4 (mπ ~ 0.8 GeV)

63 x 4 lattice, Clover fermion

11

Phase Boundaries from Maxwell Construction

Nf = 4 Wilson gauge + fermion action

Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

Maxwell construction : determine phase boundary

2

1

1 1 2 2( ) ( )

( '( ) ) 0

B B

B

F F

d F

13

Ph. Forcrand,S.Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 624 flavor (taste) staggered fermion

Phase Boundaries

UK, 2007, page 14

Three flavor case (mπ ~ 0.8 GeV)63 x 4 lattice, Clover fermion

UK, 2007, page 15

Critial Point of Nf = 3 Case

TE = 0.94(4) TC, µE = 3.01(12) TC

mπ ~ 0.8 GeV, 63 x 4 lattice

Transition density ~ 5-8 ρNMNature of phase

transition

16

Polyakov Loop

17

Quark Condensate

18

Sign Problem?

Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

UpdatePrevious results were based on accepted configurations with 16 discrete Φ’s in the WNEM and reweighting with `exact’ FT for detk with sufficient Φ’s (n=[16, 128]) so that An/A1 is less than 10-15.Updated results on NF = 3 is from accepted configurations with sufficient Φ’s for `exact’ detk.

Chemical Potential at T = 0.83 TC

Challenges for more realistic calculations

Smaller quark masses: HYP smearing, larger volume.Larger volume: • larger quark number k to maintain the same baryon

density numerically more intensive to calculate more Φ’s.

• Any number of Φ’s with the evaluation of determinant in spatial dimension for the Wilson-Clover fermion is developed by Urs Wenger (see poster by A. Alexandru).

• Acceptance rate and sign problem isospin chemical potential in HMC and A/R with detk.

†| | | | logdet ( )† †det ( ; ) det ( ) kk k D D UT T

I kk k

D D U e D D U e

† †

| |† †

† †

| | logdet ( ) logdet ( )

det ( ; ) det ( )det ( ) det ( )

k

kI kT

k

k D D U D D UT

k

D D U D D UeD D U D D U

e

SummaryCalculation with small volume and large quark mass notwithstanding, direct observation of the first order phase transition and the critical point is afforded in the canonical ensemble approach.Problem with precise numerical evaluation of detk for large k is largely overcome.Simulations at smaller quark masses and larger volumes free from acceptance rate and sign problems are essential to the full understanding of the realistic situation.

INT 2008, page 26

What Phases?

T

μ

Deconfined (QGP)

confined

• In pure gauge or Nf =3 massless case, first order finite T transition with Polyakov loop being the order parameter

Z3 symmetry breaking, confinement (disorder) to deconfinement (order) transition.

• With realistic QCD, finite T transition is a crossover. This is no symmetry breaking, no order parameter.

• Puzzle: Polyakov loop (world line of a static quark) is non-zero for grand canonical partition function due to the fermion determinant, but zero for canonical partition function for Nq = multiple of 3 which honors Z3 symmetry.

INT 2008, page 27

Polyakov Loop Puzzle K. Fukushima

P. de Forcrand ZC (T, B=3q) obeys Z3 symmetry (U4(x) -> ei2Π/3 U4(x)),

Fugacity expansion

Canonical ensemble and grand canonical ensemble

should be the same at thermodynamic limit. Suggestion: examine correlator which is Z3 invariant, instead of the Polyakov loop which is a vacuum expectation value.

2 /3 2 /3( , ) (1 ) / ( , ) 0

C

i iZ T B i i C

i

P WP e e Z T B

0 1 2( , 0)

( ))... ...

0( , 0) ... (0) (3) (6) ...GC

q i iq i

Z TGC C C C

P PW qP P P

PZ T Z Z Z

),(),,/( / VTZeVTTZ B

V

VB

TBGC

†( ) (0)P x P

Cluster Decompositon † †( ) (0) 0 (?) in ( , )CxP x P P P Z T B

Counter example at infinite volume for ZC(T,B)

To properly determine if there is a vacuum condensate, one needs to introduce a small symmetry breaking and take the infinite volume limit BEFORE taking the breaking term to zero.

INT 2008, page 29

Example: quark condensate

2 22 2

( )2 (0), V first2 1

0, 0 first a a

dmmm

V m m

0 cl him irl aim l symmetry bre 0 aking

m V

Infinite volume is important ! Polyakov loop is non-zero and the same in grand canonical and canonical ensembles at infinite volume, except at zero temperature.

Winding number expansion in canonical approach to finite density Xiangfei Meng - Lattice 2008 Williamsburg

Winding number expansion (II)For

So

The first order of winding number expansion

Here the important is that the FT integration of the first order term has analytic solution

is Bessel function of the first kind .