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N F = 3 Critical Point from Canonical Ensemble

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- 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

χQCDCollaboration:

A. Li, A. Alexandru,

KFL, and X.F. Meng

A Conjectured Phase Diagram

Canonical ensembles

Fourier transform

Canonical approach

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)

Standard HMC

Accept/Reject

Phase

Real due to

C or T, or CH

Continues Fourier transform

Useful for large k

WNEM

4

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

Winding number expansion (I)

In QCD

Tr log loop loop expansion

In particle number space

Where

Winding number expansion in canonical approach to finite densityXiangfeiMeng - Lattice 2008 Williamsburg

T

T

S

S

A0

A0

T

T

S

S

A1

A2

Observables

Polyakov loop

Baryon chemical potential

Phase

8

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

T

plasma

hadrons

coexistent

ρ

Phase diagram

Three flavors

Four flavors

?

T

plasma

hadrons

coexistent

ρ

9

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

Phase Boundaries from Maxwell Construction

Nf = 4 Wilson gauge + fermion action

Maxwell construction : determine phase boundary

11

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

Phase Boundaries

Ph. Forcrand,S.Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 62

4 flavor (taste) staggered fermion

63 x 4 lattice, Clover fermion

Three flavor case (mπ ~ 0.8 GeV)

Critial Point of Nf = 3 Case

mπ ~ 0.8 GeV, 63 x 4 lattice

TE = 0.94(4) TC,

µE = 3.01(12) TC

Transition density ~ 5-8 ρNM

Nature of phase transition

PolyakovLoop

Quark Condensate

Sign Problem?

18

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

- Previous 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

- 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.

- Calculation 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 detkforlarge 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.

- 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.

T

Deconfined (QGP)

confined

μ

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.

Cluster Decompositon

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.

Example: quark condensate

- 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 (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 .

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