Finite Density with Canonical Ensemble and the Sign Problem Finite Density Algorithm with Canonical...
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Finite Density with Canonical Ensemble and the Sign Problem • Finite Density Algorithm with Canonical Ensemble Approach • Results on N F = 4 and N F = 3 with Wilson- Clover Fermion • Nature of Phase Transition • Origin of the Sign Problem and Noise Filtering Regensburg, Oct. 19-22, 2012
Transcript of Finite Density with Canonical Ensemble and the Sign Problem Finite Density Algorithm with Canonical...
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 FilteringRegensburg, Oct. 19-22, 2012
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
T
S
T
S
T
S
T
S
A0 A0
A1 A2
5
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
page 8
Phase Diagrams
T T
ρ μ
ρq
ρh
Mixed phase
First order ?
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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
A. Li, A. Alexandru, KFL, and X. Meng, PR D 82, 054502 (2010)
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 LS
3.
• Eigenvalues of the time-reduced matrix admits
exact F.T.
/2 /2det det det[ ]i iM Q e T Ue
3
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2
1
det det ( )C S
C S
N Li N L i
ii
M Q e e
page 12
Three flavor case (mπ ~ 0.7 GeV, a~ 0.3 fm)
63 x 4 lattice, Clover fermion
Is there a sign problem?
Critial Point of Nf = 3 Case
TCP = 0.927(5) TC (~ 157 MeV) µCP = 2.60(8) TC (~ 441 MeV)
mπ ~ 0.7 GeV, 63 x 4 lattice, a ~ 0.3 fm
Transition density ~ 5-8 ρNM
A. Li, A. Alexandru, KFL, PR D84, 071503 (2011)
page 15
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
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
),(),,/(4
4
/ VTZeVTTZ k
V
Vk
TkGC
† †( ) (0) 0 (?) in ( , )CxP x P P P Z T B
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.
K. Fukushima
page 17
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
Polyakov loop is non-zero and the same in grand canonical and canonical ensembles at infinite volume, except at zero temperature.
page 18
What Phases?
T
μ
Quark Gluon Plasma (Deconfined)
Hadron (Partially
Deconfined)
0 deconfined
=0 confinedP
T=0, confined
T
gXgc
ordered
T=0, disordered
/g Te
O(3) Sigma Model
page 19
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
1
ln ( ) cumulant of ln ( )!
thnn
n
C t n C tn
page 20
Origin of Sign Problem in Canonical Approach• Finite Density -- Winding Number
Expansion
1log ( , )
2ik
kW e tr M U
1 Im
tanRe
kk
k
W
W
One loop:
Ik is the Bessel function of the first kind
1When k large and/or T low so that / 2 occurs o
Sign problem
ften
!
k
1Im is the culprit.W
Wk is the quark world line wrapping around the time boundary world loop
page 21
J/ψ from anisotropic 83 x 96 lattice
• Hadron Correlators
( ( ) )Parisi-Lepage ~ c
E p m te
SignalSign problem: ~
Noise
VfTe
page 22
(000) at t = 25 (000) at t = 40
Comparison of data with several distributions
2 20
2 21 0
(ln ) /
2
(ln( ) ) /12
1
# in bin 1, log-N= , x [0, ]
( ) 21
shifted log-N= , x [-x , ]( ) 2
x x
x x x
P eC t x
ex x
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Sub-dimensional Long Range Order of the topological charge
I. Horvath et al., Phys. Rev. D68, 114505 (2003)
Two sheets of three-dimensional coherent charges which extends over ~ 80% of space-time for each configuration and survives the continuum limit.
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Complex Distribution of C(t) for (000) Momentum
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Complex Distribution of C(t) for (222) Momentum
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Real and Imaginary P
(111) t = 25 (111) t = 40
(222) t = 25 (222) t = 40
page 27
Live with Sign Problem- signal through noise filtering
• Ansatz 1:
– then
– Sign problem when
• Ansatz 2:
Re ( ) ( ) ( )N ZP x dy P x y P y If the noise PN is symmetric and
normalized,Re
2 2 2 2 2Re Re
,
ZP P
N Z Z
x y
2 2 2Re << Z N
Re Im
Im
( ) ( ) ( ),
( ) 0, CH-theorem
ZP x dy P x y P y
dz P z
Y. Yang, KFL
page 28
Fitting with log-Normal Distribution
(222) t = 25 (222) t = 40
page 29
Fitted CZ (t) vs CRe (t) with 0.5 M configurations
Systematic error since the relative error of CRe (t) is < 1%.
page 30
Fitted CZ (t) vs CRe (t) with 5k configurations
SummaryThe 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.
Overlap Problem
3 12 0 1 2 3B
BT
BFB e
Z
Z /
0
),(),,/( / VTZeVTTZ B
V
VB
TBGC
BZ
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
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Observables Observables
Polyakov loop
Baryon chemical potential
Phase
Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg
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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
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Ph. Forcrand,S.Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 624 flavor (taste) staggered fermion
Phase Boundaries Phase Boundaries
