Autocorrelation studies in two-flavour WilsonLattice QCD using DD-HMC algorithm
Abhishek Chowdhury, Asit K. De, Sangita De Sarkar, A.Harindranath, Jyotirmoy Maiti, Santanu Mondal, Anwesa Sarkar
June 25, Lattice 2012, Cairns, Australia
• Dynamical Wilson fermion simulations at smaller quarkmasses, smaller lattice spacings and and larger lattice volumeson currently available computers have become feasible withrecent developments such as DD-HMC algorithm (Luscher).
• Measurements of autocorrelation times help us to evaluate theperformance of an algorithm in overcoming critical slowingdown.
• In addition, an accurate determination of the uncertaintyassociated with the measurement of an observable requires arealistic estimation of the autocorrelation of the observable.
• In this work we study autocorrelations of several observablesmeasured with DD-HMC algorithm using naive Wilsonfermion and gauge action.
Definitions
The normalized autocorrelation function for an observable O isdefined as, ΓO (t) = CO (t)/CO (0) where
CO (t) =1
N− t
N−t
∑r=1
(Or −〈O〉)(Or+t −〈O〉) , 〈O〉=1
N
N
∑t=1
Ot .
The integrated autocorrelation time is,
τOint =
1
2+
window
∑t=1
ΓO (t)
The error ,
δ 〈O〉=
√2τint(O)var [O]
N
For any stationary Markov chain satisfiying ergodicity and detailedbalance,
ΓO (t) = ∑n≥1
e−t/τn | ηn(O) |2
where τn =− 1lnλn
. λn’s are real eigenvalues of symmetrizedprobability transition matrix with λ0 = 1 and |λn|< 1 for n ≥ 1.τ1 =− 1
lnλ1→ exponential autocorrelation time (τexp).
• Different obsevables couples differently to the eigenmodes.
• ΓO(t) cannot be negative.
Simulation Detailsβ = 5.6
tag lattice κ block N2 Ntrj τ
B1b 243×48 0.1575 122×62 18 13128 0.5B3a , , 0.158 63×8 6 7200 0.5B3b , , 0.158 122×62 18 13646 0.5B4a , , 0.158125 63×8 8 9360 0.5B4b , , 0.158125 122×62 18 11328 0.5B5a , , 0.15825 63×8 8 6960 0.5B5b , , 0.15825 122×62 18 12820 0.5
C2 323×64 0.158 83×16 8 7576 0.5C3 , , 0.15815 83×16 8 9556 0.5C4 , , 0.15825 83×16 8 4992 0.25
β = 5.8
tag lattice κ block N2 Ntrj τ
D1 323×64 0.1543 83×16 8 9600 0.5D5 , , 0.15475 83×16 8 6820 0.25
Table: Here block, N2, Ntrj , τ refers to HMC block, step number for theforce F2, number of HMC trajectories and the Molecular Dynamicstrajectory length respectively.
Negativity of autocorrelation function for plaquette
0 40 80 120 160 200 240 280t
0
0.5
Γ (t
)
Ntrj
= 500
B5b
: L24T48, β = 5.6, κ = 0.15825
0 40 80 120 160 200 240 280t
0
0.5
Γ (t
)
Ntrj
= 1000
B5b
: L24T48, β = 5.6, κ = 0.15825
0 40 80 120 160 200 240 280t
0
0.5
Γ (t
)
Ntrj
= 5620
B5b
: L24T48, β = 5.6, κ = 0.15825
Improved Estimation of τint :Let τ∗ be the best estimate of dominant time constant. If for anobservable O all relevant time scales are smaller or of the sameorder of τ∗ then the upper bound of τint ,
τuint =
1
2+
Wu
∑t=1
ΓO (Wu) +AO(Wu)τ∗ where AO = max(ΓO(Wu),2δΓO(Wu)),
Wu chosen where autocorrelation function is still significant.
Estimate of τ∗:
τeff =t
2 ln ΓO(t/2)ΓO(t)
τexpeff = MaxO
t
2 ln ΓO(t/2)ΓO(t)
S. Schaefer et al. [ALPHA Collaboration], Nucl. Phys. B 845, 93 (2011);
S. Schaefer and F. Virotta, PoS LATTICE 2010, 042 (2010).
0 50 100 150t
0
100
200
300
400
τ eff
0 50 100 150 200 250
0
0.2
0.4
0.6
Γ(t)
0 50 100 150 200 250 300 350 4000
0.2
0.4
0.6
0.8
Γ(t)
0 50 100 150 200 250 300 350 400t
0
100
200
300
τ eff
Figure: Normalized autocorrelation function and effective autocorrelationtime for P0 (left) Q2
20 (right) for the ensemble B3b.
Autocorrelations for topological susceptibility (Q220)
0 100 200 300 400 500W/6
0
100
200
300
τ int
L24T48, β = 5.6, κ = 0.158L24T48, β = 5.6, κ = 0.158125 B
3b
B4b
0 50 100 150 2000
500
1000
1500
τ int
D1 : L32T64, β = 5.8, κ = 0.1543
D5 : L32T64, β = 5.8, κ = 0.15475
D1 D
5
W/32
Figure: Integrated autocorrelation times for topological suceptibilities(Q2
20) at β = 5.6 (a = 0.069 fm) (left) and at β = 5.8 (right) (a = 0.053fm).
τint(Q220) ↓ as κ ↑
0 100 200 300 400 500W/6
0
100
200
300
400
τ int
L24T48, β = 5.6, κ = 0.158L24T48, β = 5.6, κ = 0.158125
B3b
B4b
0 50 100 150 2000
500
1000
1500
2000
τ int
D1 : L32T64, β = 5.8, κ = 0.1543
D5 : L32T64, β = 5.8, κ = 0.15475
D1
D5
W/32
Figure: Integrated autocorrelation times and their upper bounds (τuint) for
topological suceptibilities (Q220) at β = 5.6 (a = 0.07 fm) (left) and at
β = 5.8 (right) (a = 0.055 fm).
τint(Q220) & τu
int(Q220) ↓ as κ ↑
Topological Charge Density Correlator (C(r))
0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
Tag : B : β = 5.6, Tag : D : β = 5.8
D1
B3b
t /96
Γ Q20
(t)
2
0 1 2 3 4
0
0.5
1
Γ(t)
D1, Q
20
2
D1, C(r=12.000)
B3b
, Q20
2
B3b
, C(r=12.00)
Tag B : β = 5.6, Tag D : β = 5.8
t /96
Figure: (left) Comaparison of normalized autocorrelation functions forQ2
20 at β = 5.6 and 5.8. (right) Comparison of β dependence ofnormalized autocorrelation functions for Q2
20 and C (r = 12.0).
C (r) is affected only slightly by critical slowing down.More about the properties of C(r)−→ talk by SM, Session: Chiral Symmetry
Effect of Size and Smearing
0 5 10 15 20 25 30 35 40W
0
2
4
6
8
10
12
τ int
R=1, T=1R=3, T=3R=5, T=4
D1 : L32T64, β = 5.8, κ = 0.1543
0 5 10 15W
0
1
2
3
4
5
τ int
HYP 5HYP 15HYP 25HYP 35HYP 40
C2 : L32T64, β = 5.6, κ =0.158
Figure: Integrated autocorrelation times of Wilson loops for differentsizes (left) and for different levels of HYP smearing (right).
Autocorrelation increases with increasing size and smearing level.
Pion and Nucleon Propagators
β = 5.6
tag κ τPionint τNucleon
int
B3a 0.158 99(19) 75(18)B4a 0.158125 50(9) 34(9)B5a 0.15825 40(10) 25(9)
C2 0.158 39(13) 33(17)C3 0.15815 31(15) 26(7)C4 0.15825 34(11) 18(6)
Table: Integrated autocorrelation times for pion (PP) and nucleonpropagators with wall sources.
Autocorrelation decreases with increasing κ.About low lying hadron hadrons and chiral condensate −→ talk by Asit De,
session: Chiral Symmetry
0 10 20 30 400
2
4
6
0 10 20 30 400
2
4
6
0 10 20 300
2
4
6
0 10 20 30 400
2
4
6
PP AP
PA AAτ in
t
Window
Figure: Integrated autocorrelation times for PP, AP, PA and AAcorrelators with wall source for the ensemble B3a. Measurements aredone with a gap of 24 trajectories.
P = qγ5q and
A = qγ4γ5q
q = u/d quark.
Conclusions
• Autocorrelations of topological susceptibility, pion and nucleonpropagators decrease with decreasing quark mass.
• The topological charge density correlator is affected onlyslightly by critical slowing down compared to topologicalsusceptibility.
• Increasing the size and the smearing level increases theautocorrelation of Wilson loop.
β = 5.6 MeV
κ mq mpi
0.1575 123 790
0.15755 95 684
0.158 65 562
0.158125 51 499
0.15815 49 483
0.15825 35 416
0.1583 28 378
0.1584 21 315
β = 5.8, Volume=32364 MeV
κ mq mpi
0.1543 76 600
0.15455 42 453
0.15462 31 400
0.1547 18 317
0.15475 16 275
The statistical variance of the measured value 〈O〉 is,
σ2 = 2τintσ
20
For any stationary Markov chain
ΓO (t) = ∑n≥1
(λn)t | ηn(O) |2 .
λn, are the eigenvalues of the matrix,
Tx ,y = π12
x Pxy π− 1
2y
Pxy → probability transition matrix of the Markov chain,
πx → is the stationary distribution.
ηn(O) = ∑x O(x)χn(x)π12
x where χn(x) is the eigenfunctioncorresponding to λn.
Tx ,y is positive definite. Now if the algorithm satisfies detailedbalance Tx ,y is symmetric.Then by Perron-Frobenius theorem,Tx ,y has real positive eigenvalues λn, n ≥ 0 with λ0 = 1 and|λn|< 1 for n ≥ 1.Hence,
ΓO (t) = ∑n≥1
e−t/τn | ηn(O) |2
where τn =− 1lnλn
.
τ1 =− 1lnλ1→ exponential autocorrelation time (τexp).
Q220
β κ τint τuint
5.6 0.158 247(27) 276(30)5.8 0.1543 1030(93) 1056(190)
C (r = 12.0)
β κ τint τuint
5.6 0.158 264(53) 314(76)5.8 0.1543 458(83) 850(168)
Table: Integrated autocorrelation times (τint) and their upper bounds(τu
int) for Q220 and C (r) in two β ’s.