1. Finite Density with Canonical Ensemble and the Sign Problem Finite Density Algorithm with Canonical Ensemble Approach Finite Density Algorithm with Canonical Ensemble Approach Results on N F = 4 and N F = 3 with Wilson-Clover Fermion Results on N F = 4 and N F = 3 with Wilson-Clover Fermion Nature of Phase Transition Nature of Phase Transition Origin of the Sign Problem and Noise Filtering Origin of the Sign Problem and Noise Filtering Regensburg, Oct. 19-22, 2012

2. A Conjectured Phase Diagram

3. Canonical partition function

4. T S T S T S T S A0A0 A0A0 A1A1 A2A2

5. 5 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

6. 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

8. page 8 Phase Diagrams T T ρμ ρqρq ρhρh Mixed phase First order ?

9. 9 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

10. A. Li, A. Alexandru, KFL, and X. Meng, PR D 82, 054502 (2010)

11. N F =3 A. Alexandru and U. Wenger, Phys. Rev. D83:034502 (2011) Dimension reduction in determinant calculation where the dimensions of Q and T·U are 4N C L S 3. Eigenvalues of the time-reduced matrix admits exact F.T.

12. page 12 Three flavor case (m π ~ 0.7 GeV, a~ 0.3 fm) 6 3 x 4 lattice, Clover fermion

13. Is there a sign problem?

14. Critial Point of N f = 3 Case T CP = 0.927(5) T C (~ 157 MeV) µ CP = 2.60(8) T C (~ 441 MeV) m π ~ 0.7 GeV, 6 3 x 4 lattice, a ~ 0.3 fm Transition density ~ 5-8 ρ NM A. Li, A. Alexandru, KFL, PR D84, 071503 (2011)

15. page 15 Canonical vs Grand Canonical Ensembles Canonical vs Grand Canonical Ensembles  Polyakov loop puzzle (P. de Forcrand ):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.  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.  What happens to cluster decomposition

16. 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. K. Fukushima

17. page 17 Example: quark condensate  Polyakov loop is non-zero and the same in grand canonical and canonical ensembles at infinite volume, except at zero temperature.

18. page 18 What Phases? T μ Quark Gluon Plasma (Deconfined) Hadron (Partially Deconfined) T=0, confined T g X gcgc ordered T=0, disordered O(3) Sigma Model

19. page 19 Noise and Sign Problems in Canonical Approach Noise and Sign Problems in Canonical Approach Noise problem when C (t) ≥ 0, P c is close to Noise problem when C (t) ≥ 0, P c is close to log-normal distribution [Endres, Kaplan, Lee, Nicholson, log-normal distribution [Endres, Kaplan, Lee, Nicholson, PRL 107, 201601 (2011)] PRL 107, 201601 (2011)] Sign problem when C (t) not positive definite Sign problem when C (t) not positive definite  ~ 0  ~ 0

20. page 20 Origin of Sign Problem in Canonical Approach Finite Density -- Winding Number Expansion Finite Density -- Winding Number Expansion One loop: I k is the Bessel function of the first kind W k is the quark world line wrapping around the time boundary world loop

21. page 21 J/ψ from anisotropic 8 3 x 96 lattice Hadron Correlators

22. page 22 (000) at t = 25(000) at t = 40 Comparison of data with several distributions

23. page 23 Sub-dimensional Long Range Order of the topological charge I. Horvath et al., Phys. Rev. D68, 114505 (2003) Two sheets of three-dimensional coherent charges which extends over ~ 80% of space-time for each configuration and survives the continuum limit.

24. page 24 Complex Distribution of C(t) for (000) Momentum

25. page 25 Complex Distribution of C(t) for (222) Momentum

26. page 26 Real and Imaginary P (111) t = 25 (111) t = 40 (222) t = 25(222) t = 40

27. page 27 Live with Sign Problem - signal through noise filtering Ansatz 1: Ansatz 1: – then –Sign problem when Ansatz 2: Ansatz 2: If the noise P N is symmetric and normalized, Y. Yang, KFL

28. page 28 Fitting with log-Normal Distribution (222) t = 25 (222) t = 40

29. page 29 Fitted C Z (t) vs C Re (t) with 0.5 M configurations Systematic error since the relative error of C Re (t) is < 1%.

30. page 30 Fitted C Z (t) vs C Re (t) with 5k configurations

31. Summary The first order phase transition and the critical point are observed for small volume and large quark mass in the canonical ensemble approach. Sign problem sets in quickly with larger volume. Origin of the sign problem in canonical approach. C’est la vie approach to ameliorate the sign problem with noise filitering is introduced.

32. Overlap Problem

33. 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

34. 34 ObservablesObservables Polyakov loop Baryon chemical potential Phase Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

35. 35 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

36. 36

37. 37 Ph. Forcrand,S.Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 62 4 flavor (taste) staggered fermion Phase Boundaries