Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from...

Results from combining ensembles

at several values of chemical potential

Xiao-Yong Jin

RIKEN AICS

In collaboration with

Y. Kuramashi, Y. Nakamura, S. Takeda and A. Ukawa

Mainz, Germany, Lattice 2013

July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin

Introduction

• Simulations with 4 flavors [arXiv:1307.7205] described in S. Takeda's talk

• Phase quenched simulations of grand canonical ensembles

Z||(µ) =∫

[dU ] e−Sg (U )+Nf ln|detD(µ;U )|

• (β, κ )= (1.58,0.1385) and (1.60,0.1371).

• Nt = 4 with spatial volumes from 63 to 103.

• 50,000 up to ∼ 300,000 trajectories with 1/10 measured.

• µ-reweighting using Taylor expansion of ln detD works well.

Suscep>bility�

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

!P

"=1.58

63

668688

83

103

"=1.60

0

10

20

30

40

50

60

0.1 0.12 0.14 0.16 0.18 0.2

!q

aµ

0.16 0.18 0.2 0.22 0.24

aµ

Plaqueee�

Quark&Number�

First order at β = 1.58 Weak at β = 1.60


Contents of this talk

Multi-ensemble reweighting and probability density.

Pitfalls and how to avoid them.

Zeros of the partition function with a complex µ .


Reweighting

• Probability density

Prob(U ) = 1

Z exp[−S(U )]MC−→ 1

N

N∑l=1

δ(U −Ul)

• Reweight from simulation Ss to target St using weight wt(U )

Probt(U ) = Probs(U )exp[Ss(U )− St(U )]Zs

Zt

• From multiple simulations of S1, S2, . . .

Probt(U ) =∑sps (U ) Probs (U )exp[Ss (U )− St(U )]

ZsZt

• F.–W. (1989) ⇒ Choose ps (U ) to minimize σ 2[Probt(U )] (Z1 chosen arbitrary)

Probt(U ) =(∑s ′Ns ′ Probs ′ (U )

)(∑sNs exp[St(U )− Ss (U )]

Z1

Zs

)−1Z1

Zt

• In general (ratio of Z determined by∫D[U ] Prob(U ) = 1)

Probt(U ) = 1

Zt

exp[−St(U )]MC-RW−→ wt(U )

∑conf

δ(U −Uconf)


Reweighted probability distribution

• Any quantity f (U ) with real valued 〈f 〉t for real actions

〈f 〉t =∫D[U ]f (U ) Probt(U ) =

∫D[U ]f (U )wt(U )

∑conf

δ(U −Uconf)

• Its probability density is

Probft (X) =

∫D[U ]δ

(X − f (U )

)Probt(U )

=∫D[U ]δ

(X − f (U )

)wt(U )

∑conf

δ(U −Uconf)

• With complex actions, for real partition functions, keep the real probability

〈f 〉t =∫D[U ]f (U ) Probt(U )

=∫D[U ]

(Re f (U )− Im f (U )

Imwt(U )

Rewt(U )

)Rewt(U )

∑conf

δ(U −Uconf)

• Define the probability density with real probability

Probf̃t (X) =

∫D[U ]δ

(X − f̃ (U )

)Re Probt(U )


Quark number using different ensembles

-5 0 5 10 15 20 25 30 350

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

traje

cto

ry

Nq = ∂ ln Z∂(µ/T )

µ = 0.13

µ = 0.14

µ = 0.15

µ = 0.16

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

PD

F

-5 0 5 10 15 20 25 30 350

500

1000

1500

2000

2500

3000

3500

4000

traje

cto

ry

Nq = ∂ ln Z∂(µ/T )

µ = 0.02

µ = 0.30

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

PD

F

• Reweighted to µ = 0.139

• Quite visible defect if reweighted from too far


Quark number—Compare

-5 0 5 10 15 20 25 30 350

5000

10000

15000

20000

25000

traje

cto

ry

Nq = ∂ ln Z∂(µ/T )

µ = 0.13

µ = 0.14

µ = 0.15

µ = 0.16

µ = 0.02

µ = 0.30

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

PD

F

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

-5 0 5 10 15 20 25 30 35

PD

F

Nq = ∂ ln Z∂(µ/T )

µ = 0.13, 0.14, 0.15, 0.16µ = 0.02, 0.30µ = 0.02, 0.13, 0.14, 0.15, 0.16 0.30

• Very slight change from remote ensembles

• More noise than signal

• We only use ensembles near the transition


Partition function zeros with complex µ

• Ratios of partition functions can be determined

ZaZb

=∑

confwa(Uconf)∑confwb(Uconf)

• Define normalized partition function at µ + iµ I

Znorm =Z(µ + iµ I )Z(µ)

• Use the symmetry to reduce the noise

Znorm =Z(µ + iµ I )+Z∗(µ − iµ I )

2 ReZ(µ)

• Additionally remove a factor exp〈lnw〉, which includes a large real factor

and an oscillating complex phase


Locations of the first zero, β = 1.58, κ = 0.1385

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.125 0.13 0.135 0.14 0.145 0.15 0.155 0.16

𝜇u�

𝜇

𝜇 = 0.13, 𝑉 = 63

𝜇 = 0.14, 𝑉 = 63

𝜇 = 0.15, 𝑉 = 63

𝜇 = 0.16, 𝑉 = 63

multi 𝜇, 𝑉 = 63

𝜇 = 0.13, 𝑉 = 62 × 8𝜇 = 0.14, 𝑉 = 62 × 8𝜇 = 0.15, 𝑉 = 62 × 8𝜇 = 0.16, 𝑉 = 62 × 8multi 𝜇, 𝑉 = 62 × 8𝜇 = 0.13, 𝑉 = 6 × 82

𝜇 = 0.14, 𝑉 = 6 × 82

𝜇 = 0.15, 𝑉 = 6 × 82

𝜇 = 0.16, 𝑉 = 6 × 82

multi 𝜇, 𝑉 = 6 × 82

𝜇 = 0.13, 𝑉 = 83

𝜇 = 0.14, 𝑉 = 83

𝜇 = 0.15, 𝑉 = 83


𝜇 = 0.14, 𝑉 = 103

𝜇 = 0.15, 𝑉 = 103

𝜇 = 0.16, 𝑉 = 103


Better statistics after combining multiple ensembles.


Volume scaling of µ I of first zeros

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1⁄103 1⁄83 1⁄826 1⁄628 1⁄63

𝜇𝐼

1⁄𝑉

0

0.01

0.02

0.03

0.04

0.05

0.06

1⁄103 1⁄83 1⁄826 1⁄628 1⁄63

𝜇𝐼

1⁄𝑉

β = 1.58, κ = 0.1385 β = 1.60, κ = 0.1371

Consistent with first order PT Inconsistent with first order


Estimate the confidence region in µ-reweighting

• How far can we reweight in µ? Zeros of Z(µ + iµ I ) with larger µ I?

• Complex β studied by Alves, Berg, and Sanielevici (1992)

• Introduce imaginary parameter by Fourier transform probability density

Z̃(µ,µ I ) =∫dX exp[iX

µ I

T] Probf (X) = 1

Z(µ)

∫D[U ] exp[if (U )

µ I

T] exp[−S(µ;U )]

• Taking f (U ) to be the quark number, Z̃(µ,µ I ) approximates Znorm up to the

first derivative of µ/T

• ApproximateNq distribution with Gaussian peaks, k stdev away from zero

with L i.i.d. samples, and consider one Gaussian (note: variance is additive)

|||||||||⟨

exp[Nq(∆µ + i∆µ I )/T ]⟩||||||||| ≥ k

√σ 2(| exp[Nq(∆µ + i∆µ I )/T ]|)

L

√∆µ2 + (∆µ I )2 ≤

√√√√ ln(Lk2 + 1

)σ 2(Nq)/T 2

k=1,Nl=63

−−−−−→ 0.2

• Circle around the simulation, volume factor hidden in the width of Nq• Effects of cancellations between two peaks not included


|Znorm| vs. µ , fixing µ I at the first zero

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

|Z|

µ


Z with complex µ , 63 (β = 1.58, κ = 0.1385)

|Znorm|0.50.20.1

0.050.020.01

0.0050.0020.001

0.00050.00020.0001

0.15 0.2 0.25 0.3

µ

0

0.05

0.1

0.15

0.2

µI

−π

−π2

0

π2

π


Spurious zeros within the confidence region

-1

0

1

2

3

4

5

0 0.05 0.1 0.15 0.2

|Z|

µ I

-1

0

1

2

3

4

5

0 0.05 0.1 0.15 0.2

|Z|

µI

• Plot |Znorm| vs. µ I , at the two second lowest zeros

• Statistically they are the same zero


Z with complex µ , 83, (β = 1.58, κ = 0.1385)

|Znorm|0.50.20.1

0.050.020.01

0.0050.0020.001

0.00050.00020.0001

0.15 0.2 0.25 0.3

µ

0

0.05

0.1

0.15

0.2

µI

−π

−π2

0

π2

π


Zeros of the partition function with complex µ

0

0.05

0.1

0.15

0.2

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3

µI

µ

V = 63

V = 62 × 8V = 6× 82

V = 83

V = 103


Volume scaling of µ I for the first two zeros

0

0.02

0.04

0.06

0.08

0.1

0.12

1/103 1/83 1/826 1/628 1/63

µI

1/V


Summary

• Taylor expansions of the logarithm of the fermion determinant enables us

to do µ-reweighting accurately and efficiently.

• µ-reweighting is excellent in the range of parameters we are interested in.

• Carefully combining ensembles at multiple µ gives better statistics.

• Locations of partition function zeros with complex chemical potential

form a curve starting from a finite real µ and extending to larger values

of both real and imaginary parts of the chemical potential.

• Volume dependence of the locations of zeros is interesting.

Results from combining ensemblescrlf at several values of ... fileJuly 30, 2013 Results from...

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