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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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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 .
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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)
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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 )
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
Quark number using different ensembles
-5 0 5 10 15 20 25 30 350
2000
4000
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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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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
multi 𝜇, 𝑉 = 83
𝜇 = 0.14, 𝑉 = 103
𝜇 = 0.15, 𝑉 = 103
𝜇 = 0.16, 𝑉 = 103
multi 𝜇, 𝑉 = 103
Better statistics after combining multiple ensembles.
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
|Znorm| vs. µ , fixing µ I at the first zero
-0.2
0
0.2
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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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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
July 30, 2013 Results from combining ensembles at several values of chemical potential Xiao-Yong Jin
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.