Finite Density Simulation with the Canonical Ensemble

1. Finite Density Simulation with the Canonical Ensemble Anyi Li, Xiangfei Meng, Andrei Alexandru, Keh-Fei Liu χQCD Collaboration

2. Outline • Canonical approach • Observables • Baryon chemical potential • Possible phase diagram • Conclusion and future plans Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

3. Canonical Approach Grand canonical partition function Fugacity expansion Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

4. Canonical Approach Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

5. 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) Canonical ensembles Discrete Fourier transform Standard HMC Accept/Reject Phase Continues Fourier transform Useful for large k (see X. Meng talk on Tues 3:10pm – 3:30pm) WNEM Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

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

7. Critical end point crossover Critical end point T T plasma plasma hadrons hadrons coexistent coexistent ρ ρ Phase Diagram Two flavors Four flavors ? Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

8. Phase Boundary Ph. Forcrand,S.Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 62 Maxwell construction : determine phase boundary Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

9. Baryon Chemical Potential Nf = 4 Wilson gauge + fermionaction mπ~ 1 GeV Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

10. Maxwell Construction Nf = 4 Wilson gauge + fermion action Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

11. Phase Boundary (Preliminary) Nf = 4 This work Ph. Forcrand,S.Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 62 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

12. Baryon Chemical Potential Nf = 2 Wilson gauge + fermion action Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

13. Critical end point T T plasma plasma hadrons hadrons coexistent coexistent ρ ρ Phase Diagram Two flavors Four flavors ? Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

14. Phase Diagram • “S-shape” is observed for Nf = 4 • “S-shape” is not observed down to 0.83Tc for Nf = 2 • Calculation at lower temperature is needed for two flavors Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

15. crossover Critical end point T plasma hadrons coexistent ρ Three Flavors Nf = 3 Iwasaki gauge + Clover fermionaction Lower quark mass ? Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

16. Three Flavors (Preliminary) Nf = 3 Iwasaki gauge + Clover fermionaction Lower quark mass Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

17. Conclusion and Future Plans • Continuous Fourier transform makes larger density simulation feasible • First order is observed for four flavors below Tc • First order is not observed for two flavors down to 0.83Tc • Implement improved action. • Small quark mass • Look for “S-shape” and critical end point for three flavors Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg