1. N F = 3 Critical Point from Canonical Ensemble χ QCD Collaboration: A. Li, A. Alexandru, KFL, and X.F. Meng Finite Density Algorithm with Canonical Approach and Winding Number Expansion Finite Density Algorithm with Canonical Approach and Winding Number Expansion Update on N F = 4, and 3 with Wilson-Clover Fermion Update on N F = 4, and 3 with Wilson-Clover Fermion New Algorithm Aiming at Large Volumes New Algorithm Aiming at Large Volumes

2. A Conjectured Phase Diagram

3. Canonical partition function

4. 4 Standard HMC Standard HMCAccept/RejectPhase Canonical approach Continues Fourier transform Useful for large k Continues Fourier transform Useful 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) WNEMWNEM Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg Real due to C or T, or CH

5. Xiangfei Meng - Lattice 2008 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 Tr log loop loop expansion In particle number space Where

6. T S T S T S T S A0A0 A0A0 A1A1 A2A2

8. 8 ObservablesObservables Polyakov loop Baryon chemical potential Phase Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

9. 9 Phase diagram T ρ coexistent hadrons plasma Four flavors T ρ coexistent hadrons plasma Three flavors?? Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

10. Baryon Chemical Potential for N f = 4 (m π ~ 0.8 GeV) Baryon Chemical Potential for N f = 4 (m π ~ 0.8 GeV) 6 3 x 4 lattice, Clover fermion

11. 11 Phase Boundaries from Maxwell Construction N f = 4 Wilson gauge + fermion action Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg Maxwell construction : determine phase boundary

13. 13 Ph. Forcrand,S.Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 62 4 flavor (taste) staggered fermion Phase Boundaries

14. UK, 2007, page 14 Three flavor case (m π ~ 0.8 GeV) 6 3 x 4 lattice, Clover fermion

15. UK, 2007, page 15 Critial Point of N f = 3 Case T E = 0.94(4) T C, µ E = 3.01(12) T C m π ~ 0.8 GeV, 6 3 x 4 lattice Transition density ~ 5-8 ρ NM Nature of phase transition

16. 16 Polyakov Loop

17. 17 Quark Condensate

18. 18 Sign Problem? Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

19. Update P Previous results were based on accepted configurations with 16 discrete Φ’s in the WNEM and reweighting with `exact’ FT for det k with sufficient Φ’s (n=[16, 128]) so that A n /A 1 is less than 10 -15. Updated results on N F = 3 is from accepted configurations with sufficient Φ’s for `exact’ det k.

20. Chemical Potential at T = 0.83 T C

22. 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. 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). 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 det k. Acceptance rate and sign problem  isospin chemical potential in HMC and A/R with det k.

25. Summary 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 det k 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.

26. INT 2008, page 26 What Phases? T μ Deconfined (QGP) confined In pure gauge or N f =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 Z 3 symmetry.

27. INT 2008, page 27 Polyakov Loop Puzzle K. Fukushima P. de Forcrand  Z C (T, B=3q) obeys Z 3 symmetry (U 4 (x) -> e i2Π/3 U 4 (x)),  Fugacity expansion  Canonical ensemble and grand canonical ensemble should be the same at thermodynamic limit.  Suggestion: examine correlator which is Z 3 invariant, instead of the Polyakov loop which is a vacuum expectation value.

28. Cluster Decompositon Counter example at infinite volume for Z C (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.

29. INT 2008, page 29 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.

30. Xiangfei Meng - Lattice 2008 Winding number expansion in canonical approach to finite density Xiangfei Meng - Lattice 2008 Williamsburg Winding number expansion (II) ForSo 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. is Bessel function of the first kind.