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Finite Density with Canonical Ensemble and the Sign Problem

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

  2. A Conjectured Phase Diagram

  3. Canonical partition function

  4. T T S S A0 A0 T T S S A1 A2

  5. 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 5 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

  6. 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

  7. Phase Diagrams T T First order ? ρq ρh Mixed phase ρ μ

  8. Phase Boundaries from Maxwell Construction Nf = 4 Wilson gauge + fermion action Maxwell construction : determine phase boundary 9 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

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

  10. NF =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 4NC LS3. • Eigenvalues of the time-reduced matrix admits • exact F.T.

  11. Three flavor case (mπ ~ 0.7 GeV, a~ 0.3 fm) 63 x 4 lattice, Clover fermion

  12. Is there a sign problem?

  13. Critial Point of Nf = 3 Case mπ ~ 0.7 GeV, 63 x 4 lattice, a ~ 0.3 fm TCP = 0.927(5) TC(~ 157 MeV) µCP = 2.60(8) TC (~ 441 MeV) Transition density ~ 5-8 ρNM A. Li, A. Alexandru, KFL, PR D84, 071503 (2011)

  14. 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 Z3 symmetry. • 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. • What happens to cluster decomposition

  15. K. Fukushima 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.

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

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

  18. Noise and Sign Problems in Canonical Approach • Noise problem when C (t) ≥ 0, Pc is close to log-normal distribution [Endres, Kaplan, Lee, Nicholson, PRL 107, 201601 (2011)] • Sign problem when C (t) not positive definite  <sign> ~ 0

  19. Origin of Sign Problem in Canonical Approach • Finite Density -- Winding Number Expansion Wk is the quark world line wrapping around the time boundary world loop One loop: Ik is the Bessel function of the first kind

  20. HadronCorrelators J/ψ from anisotropic 83 x 96 lattice

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

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

  23. Complex Distribution of C(t) for (000) Momentum

  24. Complex Distribution of C(t) for (222) Momentum

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

  26. Live with Sign Problem- signal through noise filtering Y. Yang, KFL • Ansatz 1: • then • Sign problem when • Ansatz 2: If the noise PN is symmetric and normalized,

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

  28. Fitted CZ (t) vsCRe (t) with 0.5 M configurations Systematic error since the relative error of CRe (t) is < 1%.

  29. Fitted CZ (t) vsCRe (t) with 5k configurations

  30. 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.

  31. Overlap Problem

  32. Baryon Chemical Potential for Nf = 4 (mπ ~ 0.8 GeV) 63 x 4 lattice, Clover fermion

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

  34. T plasma hadrons coexistent ρ Phase diagram Three flavors Four flavors ? T plasma hadrons coexistent ρ 35 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

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