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NF = 3 Critical Point from Canonical Ensemble
χQCD Collaboration: A. Li, A. Alexandru, KFL, and X.F. Meng
• Finite Density Algorithm with Canonical Approach and Winding Number Expansion
• Update on NF = 4, and 3 with Wilson-Clover Fermion
• New Algorithm Aiming at Large Volumes
A Conjectured Phase Diagram
Canonical partition function
)(det),,( 2][ UMeDUTkVZ kUS
CG
),,(2
1),,(
get we),,(),,( expansion fugacity theUsing
2
0
4
4
TTiVZedTkVZ
eTkVZTVZ
GCik
C
Vk
Vk
kT
CGC
4
Standard HMC Accept/Reject Phase
Canonical approachCanonical approach
Continues Fourier transformUseful for large k
Continues Fourier transformUseful for large k
Canonical ensembles
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, 114513 (2005)
WNEMWNEM
Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg
2 21det ( ) det ( )
2ik
k M U d e M U
Real due to
C or T, or CH
Winding number expansion in canonical approach to finite density Xiangfei Meng - Lattice 2008 Williamsburg
Winding number expansion (I)
In QCD
Tr log loop loop expansion
In particle number space
Where
T
S
T
S
T
S
T
S
A0 A0
A1 A2
8
Observables Observables
Polyakov loop
Baryon chemical potential
Phase
Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg
9
Phase diagramPhase diagram
T
ρ
coexistenthadrons
plasma
Four flavors
T
ρ
coexistenthadrons
plasma
Three flavors
??
Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg
Baryon Chemical Potential for Nf = 4 (mπ ~ 0.8 GeV)
Baryon Chemical Potential for Nf = 4 (mπ ~ 0.8 GeV)
63 x 4 lattice, Clover fermion
11
Phase Boundaries from Maxwell ConstructionPhase Boundaries from Maxwell Construction
Nf = 4 Wilson gauge + fermion action
Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg
Maxwell construction : determine phase boundary
2
1
1 1 2 2( ) ( )
( '( ) ) 0
B B
B
F F
d F
13
Ph. Forcrand,S.Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 624 flavor (taste) staggered fermion
Phase Boundaries Phase Boundaries
UK, 2007, page 14
Three flavor case (mπ ~ 0.8 GeV)63 x 4 lattice, Clover fermion
UK, 2007, page 15
Critial Point of Nf = 3 Case
TE = 0.94(4) TC, µE = 3.01(12) TC
mπ ~ 0.8 GeV, 63 x 4 lattice
Transition density ~ 5-8 ρNM
Nature of phase transition
16
Polyakov Loop
17
Quark Condensate
18
Sign Problem? Sign Problem?
Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg
Update
Previous results were based on accepted configurations with 16 discrete Φ’s in the WNEM and reweighting with `exact’ FT for detk with sufficient Φ’s (n=[16, 128]) so that An/A1 is less than 10-15.
Updated results on NF = 3 is from accepted configurations with sufficient Φ’s for `exact’ detk.
Chemical Potential at T = 0.83 TC
Challenges for more realistic calculations
Smaller quark masses: HYP smearing, larger volume.
Larger volume: • larger quark number k to maintain the same baryon
density numerically more intensive to calculate more Φ’s.
• Any number of Φ’s with the evaluation of determinant in spatial dimension for the Wilson-Clover fermion is developed by Urs Wenger (see poster by A. Alexandru).
• Acceptance rate and sign problem isospin chemical potential in HMC and A/R with detk.
†| | | |logdet ( )† †det ( ; ) det ( )
kk k
D D UT T
I kk k
D D U e D D U e
† †
| |† †
† †
| |logdet ( ) logdet ( )
det ( ; ) det ( )
det ( ) det ( )
k
kI kT
k
kD D U D D U
T
k
D D U D D Ue
D D U D D U
e
SummaryCalculation with small volume and large quark mass notwithstanding, direct observation of the first order phase transition and the critical point is afforded in the canonical ensemble approach.Problem with precise numerical evaluation of detk for large k is largely overcome.
Simulations at smaller quark masses and larger volumes free from acceptance rate and sign problems are essential to the full understanding of the realistic situation.
INT 2008, page 26
What Phases?
T
μ
Deconfined (QGP)
confined
• In pure gauge or Nf =3 massless case, first order finite T transition with Polyakov loop being the order parameter
Z3 symmetry breaking, confinement (disorder) to deconfinement (order) transition.
• With realistic QCD, finite T transition is a crossover. This is no symmetry breaking, no order parameter.
• Puzzle: 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.
INT 2008, page 27
Polyakov Loop Puzzle K. Fukushima
P. de Forcrand
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. Suggestion: examine correlator which is Z3 invariant, instead of the Polyakov loop which is a vacuum expectation value.
2 /3 2 /3( , ) (1 ) / ( , ) 0
C
i iZ T B i i C
i
P W P e e Z T B
0 1 2( , 0)
( ))... ...
0( , 0) ... (0) (3) (6) ...GC
q i iq i
Z TGC C C C
P PW qP P P
PZ T Z Z Z
),(),,/( / VTZeVTTZ B
V
VB
TBGC
†( ) (0)P x P
Cluster Decompositon
† †( ) (0) 0 (?) in ( , )CxP x P P P Z T B
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.
INT 2008, page 29
Example: quark condensate
2 22 2
( )2 (0), V first2 1
0, 0 first a a
dmm
mV m
m
0 cl him irl aim l symmetry bre 0 aking
m V
Infinite volume is important ! Polyakov loop is non-zero and the same in grand canonical and canonical ensembles at infinite volume, except at zero temperature.
Winding number expansion in canonical approach to finite density Xiangfei Meng - Lattice 2008 Williamsburg
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 .