N F = 3 Critical Point from Canonical Ensemble. Finite Density Algorithm with Canonical Approach and Winding Number Expansion Update on N F = 4, and 3 with Wilson-Clover Fermion New Algorithm Aiming at Large Volumes.
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χQCDCollaboration:
A. Li, A. Alexandru,
KFL, and X.F. Meng
A Conjectured Phase Diagram
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
4
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
T
T
S
S
A0
A0
T
T
S
S
A1
A2
Observables
Polyakov loop
Baryon chemical potential
Phase
8
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
9
Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg
Baryon Chemical Potential for Nf = 4
(mπ ~ 0.8 GeV)
63 x 4 lattice, Clover fermion
Phase Boundaries from Maxwell Construction
Nf = 4 Wilson gauge + fermion action
Maxwell construction : determine phase boundary
11
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
63 x 4 lattice, Clover fermion
Three flavor case (mπ ~ 0.8 GeV)
Critial Point of Nf = 3 Case
mπ ~ 0.8 GeV, 63 x 4 lattice
TE = 0.94(4) TC,
µE = 3.01(12) TC
Transition density ~ 5-8 ρNM
Nature of phase transition
PolyakovLoop
Quark Condensate
Sign Problem?
18
Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg
Chemical Potential at T = 0.83 TC
T
Deconfined (QGP)
confined
μ
Polyakov Loop Puzzle K. Fukushima P. de Forcrand
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
Winding number expansion (II)
For
So
The first order of winding number expansion
Here the important is that the FT integration of the first order term has analytic solution
is Bessel function of the first kind .
Winding number expansion in canonical approach to finite density Xiangfei Meng - Lattice 2008 Williamsburg