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
T T S S A0 A0 T T S S A1 A2
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
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
Phase Diagrams T T First order ? ρq ρh Mixed phase ρ μ
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
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.
Three flavor case (mπ ~ 0.7 GeV, a~ 0.3 fm) 63 x 4 lattice, Clover fermion
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)
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
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.
Example: quark condensate • Polyakov loop is non-zero and the same in grand • canonical and canonical ensembles at infinite volume, • except at zero temperature.
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
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
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
HadronCorrelators J/ψ from anisotropic 83 x 96 lattice
Comparison of data with several distributions (000) at t = 25 (000) at t = 40
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)
Real and Imaginary P (111) t = 40 (111) t = 25 (222) t = 25 (222) t = 40
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,
Fitting with log-Normal Distribution (222) t = 25 (222) t = 40
Fitted CZ (t) vsCRe (t) with 0.5 M configurations Systematic error since the relative error of CRe (t) is < 1%.
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.
Baryon Chemical Potential for Nf = 4 (mπ ~ 0.8 GeV) 63 x 4 lattice, Clover fermion
Observables Polyakov loop Baryon chemical potential Phase 34 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 35 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