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 Canonical ensembles Discrete 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, (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. ? Phase Diagram Two flavors Four flavors T ρ T ρ Critical end point
coexistent
hadrons
plasma
crossover T ρ
Critical end point
coexistent
hadrons
plasma ?
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 + fermion action 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. ? Phase Diagram Two flavors Four flavors T ρ T ρ coexistent hadrons
plasma T ρ
Critical end point
coexistent
hadrons
plasma ?
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. Three Flavors
Nf = 3 Iwasaki gauge + Clover fermion action Lower quark mass ? T ρ
Critical end point
coexistent
hadrons
plasma
crossover
Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

16. Three Flavors (Preliminary)
Nf = 3 Iwasaki gauge + Clover fermion action 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