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.