Autocorrelation studies in two-flavour Wilson Lattice QCD using DD-HMC algorithm Abhishek Chowdhury, Asit K. De, Sangita De Sarkar, A. Harindranath, Jyotirmoy Maiti, Santanu Mondal, Anwesa Sarkar June 25, Lattice 212, Cairns, Australia
Dynamical Wilson fermion simulations at smaller quark masses, smaller lattice spacings and and larger lattice volumes on currently available computers have become feasible with recent developments such as DD-HMC algorithm (Luscher). Measurements of autocorrelation times help us to evaluate the performance of an algorithm in overcoming critical slowing down. In addition, an accurate determination of the uncertainty associated with the measurement of an observable requires a realistic estimation of the autocorrelation of the observable. In this work we study autocorrelations of several observables measured with DD-HMC algorithm using naive Wilson fermion and gauge action.
Definitions The normalized autocorrelation function for an observable O is defined as, Γ O (t) = C O (t)/c O () where C O (t) = 1 N t N t r=1 The integrated autocorrelation time is, (O r O )(O r+t O ), O = 1 N τint O = 1 window 2 + Γ O (t) t=1 N O t. t=1
The error, δ O = 2τint (O)var[O] For any stationary Markov chain satisfiying ergodicity and detailed balance, N Γ O (t) = e t/τ n η n (O) 2 n 1 where τ n = 1 lnλ n. λ n s are real eigenvalues of symmetrized probability transition matrix with λ = 1 and λ n < 1 for n 1. τ 1 = 1 lnλ 1 exponential autocorrelation time (τ exp ). Different obsevables couples differently to the eigenmodes. Γ O (t) cannot be negative.
Simulation Details β = 5.6 tag lattice κ block N 2 N trj τ B 1b 24 3 48.1575 12 2 6 2 18 13128.5 B 3a,,.158 6 3 8 6 72.5 B 3b,,.158 12 2 6 2 18 13646.5 B 4a,,.158125 6 3 8 8 936.5 B 4b,,.158125 12 2 6 2 18 11328.5 B 5a,,.15825 6 3 8 8 696.5 B 5b,,.15825 12 2 6 2 18 1282.5 C 2 32 3 64.158 8 3 16 8 7576.5 C 3,,.15815 8 3 16 8 9556.5 C 4,,.15825 8 3 16 8 4992.25 β = 5.8 tag lattice κ block N 2 N trj τ D 1 32 3 64.1543 8 3 16 8 96.5 D 5,,.15475 8 3 16 8 682.25 Table: Here block, N 2, N trj, τ refers to HMC block, step number for the force F 2, number of HMC trajectories and the Molecular Dynamics trajectory length respectively.
Negativity of autocorrelation function for plaquette B 5b : L24T48, β = 5.6, κ =.15825 N trj = 5 B 5b : L24T48, β = 5.6, κ =.15825 N trj = 1.5.5 Γ (t) Γ (t) 4 8 12 16 2 24 28 t 4 8 12 16 2 24 28 t B 5b : L24T48, β = 5.6, κ =.15825 N trj = 562.5 Γ (t) 4 8 12 16 2 24 28 t
S. Schaefer et al. [ALPHA Collaboration], Nucl. Phys. B 845, 93 (211); S. Schaefer and F. Virotta, PoS LATTICE 21, 42 (21). Improved Estimation of τ int : Let τ be the best estimate of dominant time constant. If for an observable O all relevant time scales are smaller or of the same order of τ then the upper bound of τ int, τint u = 1 W u 2 + Γ O (W u ) + A O (W u )τ where A O = max(γ O (W u ),2δΓ O (W u )), t=1 W u chosen where autocorrelation function is still significant. Estimate of τ : τ exp eff τ eff = = Max O t 2ln ΓO (t/2) Γ O (t) t 2ln ΓO (t/2) Γ O (t)
Γ(t) τ eff.6.4.2 5 1 15 2 25 4 3 2 1 5 1 15 t Γ(t) τ eff.8.6.4.2 5 1 15 2 25 3 35 4 3 2 1 5 1 15 2 25 3 35 4 t Figure: Normalized autocorrelation function and effective autocorrelation time for P (left) Q 2 2 (right) for the ensemble B 3b.
Autocorrelations for topological susceptibility (Q 2 2 ) 3 L24T48, β = 5.6, κ =.158 L24T48, β = 5.6, κ =.158125 B 3b 15 D 1 : L32T64, β = 5.8, κ =.1543 D 5 : L32T64, β = 5.8, κ =.15475 1 2 τ int τ int D 1 D 5 1 5 B 4b 1 2 3 4 5 W/6 5 1 15 2 W/32 Figure: Integrated autocorrelation times for topological suceptibilities (Q2 2 ) at β = 5.6 (a =.69 fm) (left) and at β = 5.8 (right) (a =.53 fm). τ int (Q 2 2 ) as κ
4 3 L24T48, β = 5.6, κ =.158 L24T48, β = 5.6, κ =.158125 B 3b 2 15 D 1 : L32T64, β = 5.8, κ =.1543 D 5 : L32T64, β = 5.8, κ =.15475 D 1 τ int 2 τ int 1 1 5 B 4b 1 2 3 4 5 W/6 D 5 5 1 W/32 15 2 Figure: Integrated autocorrelation times and their upper bounds (τint u ) for topological suceptibilities (Q2 2 ) at β = 5.6 (a =.7 fm) (left) and at β = 5.8 (right) (a =.55 fm). τ int (Q 2 2 ) & τu int (Q2 2 ) as κ
Topological Charge Density Correlator (C(r)) 1.8 Tag : B : β = 5.6, Tag : D : β = 5.8 D 1 1 Tag B : β = 5.6, Tag D : β = 5.8 Γ 2 Q2 (t).6.4 B 3b.2 1 2 t/96 3 4 Γ(t).5 D 1, Q 2 2 D 1, C(r=12.) B 3b, Q 2 2 B 3b, C(r=12.) 1 2 3 4 t /96 Figure: (left) Comaparison of normalized autocorrelation functions for Q2 2 at β = 5.6 and 5.8. (right) Comparison of β dependence of normalized autocorrelation functions for Q2 2 and C(r = 12.). 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 τ int 12 1 8 6 4 D 1 : L32T64, β = 5.8, κ =.1543 R=1, T=1 R=3, T=3 R=5, T=4 τ int 5 4 3 2 C 2 : L32T64, β = 5.6, κ =.158 HYP 5 HYP 15 HYP 25 HYP 35 HYP 4 2 1 5 1 15 2 25 3 35 4 W 5 1 15 W Figure: Integrated autocorrelation times of Wilson loops for different sizes (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 τint Pion τ Nucleon int B 3a.158 99(19) 75(18) B 4a.158125 5(9) 34(9) B 5a.15825 4(1) 25(9) C 2.158 39(13) 33(17) C 3.15815 31(15) 26(7) C 4.15825 34(11) 18(6) Table: Integrated autocorrelation times for pion (PP) and nucleon propagators with wall sources. Autocorrelation decreases with increasing κ. About low lying hadron hadrons and chiral condensate talk by Asit De, session: Chiral Symmetry
6 4 PP 6 4 AP 2 2 τ int 1 2 3 4 6 4 PA 1 2 3 4 6 4 AA 2 2 1 2 3 1 2 3 4 Window Figure: Integrated autocorrelation times for PP, AP, PA and AA correlators with wall source for the ensemble B 3a. Measurements are done with a gap of 24 trajectories. P = qγ 5 q and A = qγ 4 γ 5 q q = u/d quark.
Conclusions Autocorrelations of topological susceptibility, pion and nucleon propagators decrease with decreasing quark mass. The topological charge density correlator is affected only slightly by critical slowing down compared to topological susceptibility. Increasing the size and the smearing level increases the autocorrelation of Wilson loop.
β = 5.6 MeV κ m q m pi.1575 123 79.15755 95 684.158 65 562.158125 51 499.15815 49 483.15825 35 416.1583 28 378.1584 21 315 β = 5.8, Volume=32 3 64 MeV κ m q m pi.1543 76 6.15455 42 453.15462 31 4.1547 18 317.15475 16 275
The statistical variance of the measured value O is, σ 2 = 2τ int σ 2 For any stationary Markov chain Γ O (t) = (λ n ) t η n (O) 2. n 1 λ n, are the eigenvalues of the matrix, 2 T x,y = π 1 x P xy π 2 1 y P xy probability transition matrix of the Markov chain, π x is the stationary distribution. η n (O) = x O(x)χ n (x)π 1 2 x corresponding to λ n. where χ n (x) is the eigenfunction
T x,y is positive definite. Now if the algorithm satisfies detailed balance T x,y is symmetric. Then by Perron-Frobenius theorem, T x,y has real positive eigenvalues λ n, n with λ = 1 and λ n < 1 for n 1. Hence, Γ O (t) = e t/τ n η n (O) 2 n 1 where τ n = 1 lnλ n. τ 1 = 1 lnλ 1 exponential autocorrelation time (τ exp ).
Q 2 2 β κ τ int τ u int 5.6.158 247(27) 276(3) 5.8.1543 13(93) 156(19) C(r = 12.) β κ τ int τ u int 5.6.158 264(53) 314(76) 5.8.1543 458(83) 85(168) Table: Integrated autocorrelation times (τ int ) and their upper bounds (τint u ) for Q2 2 and C(r) in two β s.