MULTILEVEL INTEGRATION FOR FERMIONS II Marco C, Leonardo Giusti, - PowerPoint PPT Presentation

D OMAIN DECOMPOSITION AND MULTILEVEL INTEGRATION FOR FERMIONS II Marco Cè, Leonardo Giusti, Stefan Schaefer Scuola Normale Superiore, Pisa & INFN, Sezione di Pisa Lattice 2016, July 27th, 2016

Case I: disconnected correlation functions � � � � [ tr γ 5 D − 1 ( x , x )][ tr γ 5 D − 1 ( y , y )] x y = Factorization: use quark propagator factorization, case I (sink close to source) see talk by Schaefer ⇒ disconnected two point function splits in four contributions � � [ tr γ 5 D − 1 ( x , x )][ tr γ 5 D − 1 ( y , y )] � � � [ tr γ 5 D − 1 Γ ( x , x )][ tr γ 5 D − 1 � y = Γ ∗ ( y , y )] = x � � x [ tr γ 5 D − 1 Γ ( x , x )][ tr γ 5 δ D − 1 � � + Γ ∗ ( y , y )] + y � � x [ tr γ 5 δ D − 1 Γ ( x , x )][ tr γ 5 D − 1 + � Γ ∗ ( y , y )] � + y � � [ tr γ 5 δ D − 1 Γ ( x , x )][ tr γ 5 δ D − 1 x � � + Γ ∗ ( y , y )] + y 1 / 19

Numerical test Wilson plaquette action β = 6 ⇒ a ≈ 0 . 093 fm Wilson fermions κ = 0 . 1560 ⇒ m π ≈ 450 MeV 64 × 24 3 lattice with open boundary conditions in time Two-level Monte Carlo: n 0 = 200 level- 0 configurations domain decomposition in two thick time slices n 1 = 100 level- 1 configurations Estimation of tr γ 5 D − 1 : Stochastic volume sources Hopping parameter expansion to reduce UV noise Note: to be computed only 2 / 19

Contributions to propagator: factorized � [ tr γ 5 D − 1 Γ ( x , x )][ tr γ 5 D − 1 Γ ∗ ( y , y )] � + � [ tr γ 5 D − 1 Γ ( x , x )][ tr γ 5 δ D − 1 Γ ∗ ( y , y )] � + � [ tr γ 5 δ D − 1 Γ ( x , x )][ tr γ 5 D − 1 Γ ∗ ( y , y )] � + � [ tr γ 5 δ D − 1 Γ ( x , x )][ tr γ 5 δ D − 1 Γ ∗ ( y , y )] � Factorized contribution ⇒ independent averaging over level- 1 configs 10 − 2 10 − 2 C mlv δ ( f ) P d Pd n 1 = 1 � C ( f ) � � � � 10 − 3 10 − 3 P d � n 1 = 10 n 1 = 100 10 − 4 10 − 4 10 − 5 10 − 5 10 − 6 10 − 6 0 10 20 30 40 50 60 0 10 20 30 40 50 60 | y 0 − x 0 | | y 0 − x 0 | first term carries almost no signal, only noise tr γ 5 D − 1 � � Note that Γ ( x , x )] = 0 Multilevel works at full potentiality to reduce the variance 3 / 19

Contributions to propagator: correction � [ tr γ 5 D − 1 Γ ( x , x )][ tr γ 5 D − 1 Γ ∗ ( y , y )] � + � [ tr γ 5 D − 1 Γ ( x , x )][ tr γ 5 δ D − 1 Γ ∗ ( y , y )] � + � [ tr γ 5 δ D − 1 Γ ( x , x )][ tr γ 5 D − 1 Γ ∗ ( y , y )] � + � [ tr γ 5 δ D − 1 Γ ( x , x )][ tr γ 5 δ D − 1 Γ ∗ ( y , y )] � | − | | − | 10 − 2 10 − 2 C mlv δ ( r 1) P d Pd C ( r 1 ) n 1 = 1 10 − 3 P d 10 − 3 n 1 = 10 n 1 = 100 10 − 4 10 − 4 10 − 5 10 − 5 10 − 6 10 − 6 0 10 20 30 40 50 60 0 10 20 30 40 50 60 | y 0 − x 0 | | y 0 − x 0 | 10 − 2 10 − 2 C mlv δ ( r 2) P d Pd C ( r 2 ) n 1 = 1 10 − 3 P d 10 − 3 n 1 = 10 n 1 = 100 10 − 4 10 − 4 10 − 5 10 − 5 10 − 6 10 − 6 0 10 20 30 40 50 60 0 10 20 30 40 50 60 | y 0 − x 0 | | y 0 − x 0 | 4 / 19

Summary: disconnected 10 − 2 10 − 2 C P d δ P d C mlv δ mlv P d P d 10 − 3 10 − 3 10 − 4 10 − 4 10 − 5 10 − 5 10 − 6 10 − 6 0 10 20 30 40 50 60 0 10 20 30 40 50 60 | y 0 − x 0 | | y 0 − x 0 | short-distance: error dominated by the first correction long-distance: error dominated by the factorized contribution ⇒ Factorization and multilevel integration works error decreases exponentially maintain signal for additional 1 fm 5 / 19

Case II: connected correlation functions meson propagator baryon propagator � � � � Factorization less obvious. Two steps: factorization of the quark propagator, case II (sink far from source) 1 see talk by Schaefer y ≈ − x y + O e M π ∆ � x � D − 1 ( x , y ) − D − 1 Γ ( y , · ) D ∂ Γ ∗ D − 1 Γ ( · , x ) ¯ factorization of the hadron two-point function 2 With the factorized propagator: multilevel integration 3 6 / 19

Numerical test Wilson plaquette action β = 6 ⇒ a ≈ 0 . 093 fm Wilson fermions κ = 0 . 1560 ⇒ m π ≈ 450 MeV 64 × 24 3 lattice with open boundary conditions in time Test of quark propagator factorization: stochastic sources on time slice x 0 = 4 a first cut at x 0 = 24 a ∆ = 8 a , 12 a and 16 a ⇒ second cut at 24 a − ∆ Contraction of two or three quark lines: pseudoscalar propagator nucleon propagator x =24 x =24 0 0 Δ Δ 7 / 19

Meson Pseudoscalar two-point function x =24 0 10 − 2 Δ 10 − 3 10 − 4 10 − 5 C P c 10 − 6 C (0) P c 10 − 7 C (1) P c − C (0) P c 10 − 8 C (2) P c − C (1) P c 10 − 9 C ( rest ) P c 10 − 10 0 10 20 30 40 50 60 y 0 � � � � C P c = C ( 0 ) C ( 1 ) P c − C ( 0 ) C ( 2 ) P c − C ( 1 ) + C ( rest ) P c + + P c P c P c C ( 0 ) : ∆ = 8 a C ( 1 ) : ∆ = 12 a C ( 2 ) : ∆ = 16 a 8 / 19

Baryon Nucleon two-point function x =24 10 − 6 0 Δ C N 10 − 7 C (0) 10 − 8 N C (1) − C (0) 10 − 9 N N 10 − 10 C (2) − C (1) N N 10 − 11 C ( rest ) N 10 − 12 10 − 13 10 − 14 10 − 15 10 − 16 0 10 20 30 40 50 60 y 0 � � � � C N = C ( 0 ) C ( 1 ) N − C ( 0 ) C ( 2 ) N − C ( 1 ) + C ( rest ) N + + N N N C ( 0 ) : ∆ = 8 a C ( 1 ) : ∆ = 12 a C ( 2 ) : ∆ = 16 a 9 / 19

Factorized propagator Factorization of propagator in principle works small ∆ already gives excellent approximation can be improved by hierarchy of ∆ i ⇒ potential for multi-level Problem: Natural building blocks have two or three propagator on surface ( 6 V 3 ) 2 or ( 6 V 3 ) 3 complex numbers ⇒ too much to be saved to disk 10 / 19

Projection of the propagator y x Cut the fermion line with P : S ( y , x ) = − D − 1 Γ D ∂ Γ ∗ · P · D − 1 Γ ( x , y ) ¯ P projects on lower dimensional space N � ψ i ψ † P = i i = 1 Reduce memory of building block ( 6 V 3 ) 2 → N 2 Several possibilities: stochastic sources ⇒ does not seem to work, large N required local deflation subspace eigenmodes of block Dirac operator 11 / 19

Projections: deflation subspace Pseudoscalar two-point function with P : deflation subspace at x 0 / a = 24 10 − 2 10 − 2 10 − 3 10 − 3 10 − 4 10 − 4 10 − 5 10 − 5 C P c C P c 10 − 6 10 − 6 C (0) C (0) ˜ P c P c 10 − 7 C (1) C (0) 10 − 7 C (1) P c − C (0) ˜ P c − ˜ P c P c 10 − 8 C (2) C (1) 10 − 8 C (2) P c − C (1) ˜ P c − ˜ P c P c 10 − 9 C ( rest ) 10 − 9 C ( rest ) ˜ P c P c 10 − 10 10 − 10 0 10 20 30 40 50 60 0 10 20 30 40 50 60 y 0 y 0 Deflation subspace: Factorization only N s = 60 block modes, 4 4 blocks ∆ / a = 8 , 12 , 16 Lüscher ‘07 ⇒ Virtually no difference visible 12 / 19

Projections: block eigenvectors subspace Pseudoscalar two-point function with P : eigenvectors of Dirac operator restricted to x 0 ∈ [ 24 − ∆ , 24 + ∆] 10 − 2 10 − 2 10 − 3 10 − 3 10 − 4 10 − 4 10 − 5 10 − 5 C P c C P c 10 − 6 C (0) ˜ 10 − 6 C (0) P c P c 10 − 7 C (1) ˜ P c − ˜ C (0) 10 − 7 C (1) P c − C (0) P c P c 10 − 8 C (2) ˜ P c − ˜ C (1) 10 − 8 C (2) P c − C (1) P c P c 10 − 9 C ( rest ) ˜ 10 − 9 C ( rest ) P c P c 10 − 10 10 − 10 0 10 20 30 40 50 60 0 10 20 30 40 50 60 y 0 y 0 Block eigenvectors subspace: Factorization only N ev = 120 vectors ∆ / a = 8 , 12 , 16 ⇒ Virtually no difference visible 13 / 19

Projections: deflation subspace Nucleon two-point function with P : deflation subspace at x 0 / a = 24 10 − 6 10 − 6 C N C N 10 − 7 10 − 7 C (0) C (0) ˜ 10 − 8 10 − 8 N N C (1) − C (0) C (1) C (0) 10 − 9 ˜ − ˜ 10 − 9 N N N N 10 − 10 C (2) − C (1) 10 − 10 C (2) C (1) ˜ − ˜ N N N N 10 − 11 10 − 11 C ( rest ) C ( rest ) ˜ N N 10 − 12 10 − 12 10 − 13 10 − 13 10 − 14 10 − 14 10 − 15 10 − 15 10 − 16 10 − 16 0 10 20 30 40 50 60 0 10 20 30 40 50 60 y 0 y 0 Deflation subspace: Factorization only N s = 60 block modes, 4 4 blocks ∆ / a = 8 , 12 , 16 Lüscher ‘07 14 / 19

Projections: block eigenvectors subspace Nucleon two-point function with P : eigenvectors of Dirac operator restricted to x 0 ∈ [ 24 − ∆ , 24 + ∆] 10 − 6 10 − 6 C N 10 − 7 C N 10 − 7 C (0) C (0) ˜ 10 − 8 10 − 8 N N C (1) − C (0) 10 − 9 C (1) ˜ − ˜ C (0) 10 − 9 N N N N 10 − 10 C (2) − C (1) 10 − 10 C (2) C (1) ˜ − ˜ N N N N 10 − 11 10 − 11 C ( rest ) C ( rest ) ˜ N N 10 − 12 10 − 12 10 − 13 10 − 13 10 − 14 10 − 14 10 − 15 10 − 15 10 − 16 10 − 16 0 10 20 30 40 50 60 0 10 20 30 40 50 60 y 0 y 0 Block eigenvectors subspace: Factorization only N ev = 120 vectors ∆ / a = 8 , 12 , 16 ⇒ Surprise: also the nucleon can be saturated by low-modes 15 / 19

Summary of factorization Factorization of pion and nucleon two-point functions in principle possible. Two approximations involved: approximate factorization of propagator projection to low-mode subspace Surprise: the nucleon propagator is saturated by low modes! ⇒ Multilevel in quenched is possible 16 / 19