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 ?

9

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

34

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

page 23

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.

page 24

Complex Distribution of C(t) for (000) Momentum

page 25

Complex Distribution of C(t) for (222) Momentum

page 26

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

34

Observables Observables

Polyakov loop

Baryon chemical potential

Phase

Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

35

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

36

37

Ph. Forcrand,S.Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 624 flavor (taste) staggered fermion

Phase Boundaries Phase Boundaries