Comparison of different definitions of the topological charge: PART - PowerPoint PPT Presentation

Comparison of different definitions of the topological charge: PART II ———— Andreas Athenodorou ∗ University of Cyprus ———— July 27, 2016 Lattice 2016 Southampton, UK ▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻ ∗ With C. Alexandrou, K. Cichy, A. Dromard, E. G-Ramos, K. Jansen, K. Ottnad, C. Urbach, U. Wenger and F. Zimmermann. based on [arXiv:1411.1205] and [arXiv:1509.04259] and a forthcoming paper.

Preface U Several definitions of the topological charge: f fermionic (Index, Spectral flow, Spectral Projectors). g gluonic with UV fluctuations removed via smoothing (gradient flow, cooling, smearing,...). ? How are these definitions numerically related? U The gradient flow provides a well defined smoothing scheme with good renormalizability properties. M. L¨ uscher [arXiv:1006.4518] ! The gradient flow is numerically equivalent to cooling! C. Bonati and M. D’Elia [arXiv:1401.2441] and C. Alexandrou, AA and K. Jansen, [arXiv:1509.0425] ? Can this be applied to other smoothing schemes? R Comparison of different definitions presented by Krzysztof Cichy in LATTICE 2014 . . . K. Cichy et. al , [arXiv:1411.1205] ! Most definitions are highly correlated. ! The topological susceptibilities are in the same region.

Overview from Lattice 2014 Continuation of Krzysztof Cichy’s talk given in Lattice 2014: HYP1 0.1 s=0 HYP5 impr. impr. s=0 cool10 impr. APE30 cool30 cool30 flow time: HYP1 noSmear HYP1 HYP5 M 2 =0.001 HYP10 t 0 2t 0 3t 0 naive s=0.75 naive s=0.4 s=0 s=0.5 APE10 cool10 0.08 HYP30 cool30 cool10cool30 APE10APE30 cool10 M 2 =0.0015 noSmear naive M 2 =0.0004 s=0 noSmear a χ 1/4 0.06 M 2 =0.00003555 0.04 index of overlap impr. FT HYP spectral flow impr./naive FT APE 0.02 spectral projectors impr./naive FT impr. cooling field theor. GF impr./naive FT basic cooling impr. FT noSmear 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 definition ID Using N f = 2 twisted mass configuration with: β = 3 . 90, a ≃ 0 . 085fm, r 0 /a = 5 . 35(4), m π ≃ 340 MeV, m π L = 2 . 5, L/a = 16

Overview from Lattice 2014 Continuation of Krzysztof Cichy’s talk given in Lattice 2014: HYP1 0.1 s=0 HYP5 impr. impr. s=0 cool10 impr. APE30 cool30 cool30 flow time: HYP1 noSmear HYP1 HYP5 M 2 =0.001 HYP10 t 0 2t 0 3t 0 naive s=0.75 naive s=0.4 s=0 s=0.5 APE10 cool10 0.08 HYP30 cool30 cool10cool30 APE10APE30 cool10 M 2 =0.0015 noSmear naive M 2 =0.0004 s=0 noSmear a χ 1/4 0.06 M 2 =0.00003555 0.04 index of overlap impr. FT HYP spectral flow impr./naive FT APE 0.02 spectral projectors impr./naive FT impr. cooling field theor. GF impr./naive FT basic cooling impr. FT noSmear 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 definition ID ✻ Using N f = 2 twisted mass configuration with: ▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ β = 3 . 90, a ≃ 0 . 085fm, r 0 /a = 5 . 35(4), m π ≃ 340 MeV, m π L = 2 . 5, L/a = 16

Details of the Topological Charge Comparison f Index definition with different steps of HYP smearing. M. F. Atiyah and I. M. Singer, Annals Math. 93 (1971) 139149 f Spectral-flow with different steps of HYP smearing. S. Itoh, Y. Iwasaki and T. Yoshie, Phys. Rev. D 36 (1987) 527 Spectral projectors with different cutoffs M 2 . f L. Giusti and M. L¨ uscher, JHEP 0903 (2009) 013 and M. L¨ uscher and F. Palombi, JHEP 1009 (2010) 110 g The Wilson flow (also gradient flow with different actions). M. L¨ uscher, JHEP 1008 (2010) 071 g Cooling with the Wilson plaquette action (also tlSym and Iwasaki). M. Teper, Phys. Lett. B 162 (1985) 357. g APE smearing with α APE = 0 . 4 , 0 . 5 , 0 . 6. M. Albanese et al. [APE Collaboration], Phys. Lett. B 192 (1987) 163. g Stout smearing with ρ st = 0 . 01 , 0 . 05 , 0 . 1. C. Morningstar and M. J. Peardon, Phys. Rev. D69 (2004) 054501 g HYP smearing with α HYP1 = 0 . 75 , α HYP2 = 0 . 6 α HYP3 = 0 . 3. A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (2001) 034504

Details of the Topological Charge Comparison f Index definition with different steps of HYP smearing. M. F. Atiyah and I. M. Singer, Annals Math. 93 (1971) 139149 f Spectral-flow with different steps of HYP smearing. S. Itoh, Y. Iwasaki and T. Yoshie, Phys. Rev. D 36 (1987) 527 Spectral projectors with different cutoffs M 2 . f L. Giusti and M. L¨ uscher, JHEP 0903 (2009) 013 and M. L¨ uscher and F. Palombi, JHEP 1009 (2010) 110 g The Wilson flow (also gradient flow with different actions). M. L¨ uscher, JHEP 1008 (2010) 071 g Cooling with the Wilson plaquette action (also tlSym and Iwasaki). M. Teper, Phys. Lett. B 162 (1985) 357. g APE smearing with α APE = 0 . 4 , 0 . 5 , 0 . 6. M. Albanese et al. [APE Collaboration], Phys. Lett. B 192 (1987) 163. g Stout smearing with ρ st = 0 . 01 , 0 . 05 , 0 . 1. C. Morningstar and M. J. Peardon, Phys. Rev. D69 (2004) 054501 g HYP smearing with α HYP1 = 0 . 75 , α HYP2 = 0 . 6 α HYP3 = 0 . 3. A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (2001) 034504

Field Theoretic Definition of the Topological Charge g Topological charge can be defined as: � 1 d 4 x q ( x ) , Q = with q ( x ) = 32 π 2 ǫ µνρσ Tr { F µν F ρσ } . g Discretizations of q ( x ) on the lattice: R Plaquette � � 1 � � q plaq C plaq C plaq C plaq ( x ) = 32 π 2 ǫ µνρσ Tr , with ( x ) = Im . µν ρσ µν L R Clover � � 1 µν ( x ) = 1 � � q clov C clov µν C clov C clov ( x ) = 32 π 2 ǫ µνρσ Tr , with 4Im . ρσ L R Improved   µν ( x ) = 1   q imp ( x ) = b 0 q clov ( x ) + b 1 q rect C rect ( x ) , with 8 Im +  .   L L L    g Smoothing...

Field Theoretic Definition of the Topological Charge g Topological charge can be defined as: � 1 d 4 x q ( x ) , Q = with q ( x ) = 32 π 2 ǫ µνρσ Tr { F µν F ρσ } . g Discretizations of q ( x ) on the lattice: R Plaquette � � 1 � � q plaq C plaq C plaq C plaq ( x ) = 32 π 2 ǫ µνρσ Tr , with ( x ) = Im . µν ρσ µν L U Clover � � 1 µν ( x ) = 1 � � q clov C clov µν C clov C clov ( x ) = 32 π 2 ǫ µνρσ Tr , with 4Im . ρσ L R Improved   µν ( x ) = 1   q imp ( x ) = b 0 q clov ( x ) + b 1 q rect C rect ( x ) , with 8 Im +  .   L L L    g Smoothing...

Example: The Wilson flow Vs. Cooling Gradient Flow R Solution of the evolution equations: ˙ − g 2 V µ ( x, τ ) = 0 [ ∂ x,µ S G ( V ( τ ))] V µ ( x, τ ) V µ ( x, 0) = U µ ( x ) , R With link derivative defined as: � � T a d e isY a U � � � ∂ x,µ S G ( U ) = i d s S G � � � a s =0 � T a ∂ ( a ) ≡ i x,µ S G ( U ) , a R Total gradient flow time: τ R Reference flow time t 0 such that t 2 � E ( t ) �| t = t 0 = 0 . 3 with t = a 2 τ and E ( t ) = − 1 � x Tr { F µν ( x, t ) F µν ( x, t ) } 2 V Cooling R Cooling U µ ( x ) ∈ SU ( N ): U old ( x ) → U new ( x ) with µ µ P ( U ) ∝ e (lim β →∞ β 1 N ReTr X µ † U µ ) . R Choose a U new ( x ) that maximizes: µ ReTr { U new ( x ) X † µ ( x ) } . µ R One full cooling iteration n c = 1

Perturbative expansion of links R A link variable which has been smoothed can been written as: µ ( x, j sm ) T a . � u a U µ ( x, j sm ) ≃ 1 1 + i a R Simple staples are written as: per space-time slice, thus. µ ( x, j sm ) T a . � w a X µ ( x, j sm ) ≃ 6 · 1 1 + i a R For the Wilson flow with Ω µ ( x ) = U µ ( x ) X † µ ( x ) 0 ∂ x,µ S G ( U )( x ) = 1 − 1 � � � � g 2 Ω µ ( x ) − Ω † Ω µ ( x ) − Ω † µ ( x ) 6 Tr µ ( x ) . 2 where T a . g 2 � 6 u a µ ( x, τ ) − w a � � 0 ∂ x,µ S G ( U ) ≃ i µ ( x, τ ) a

Perturbative expansion of links R A link variable which has been smoothed can been written as: µ ( x, j sm ) T a . � u a U µ ( x, j sm ) ≃ 1 1 + i a R Simple staples are written as: per space-time slice, thus. µ ( x, j sm ) T a . � w a X µ ( x, j sm ) ≃ 6 · 1 1 + i a R For the Wilson flow with Ω µ ( x ) = U µ ( x ) X † µ ( x ) 0 ∂ x,µ S G ( U )( x ) = 1 − 1 � � � � g 2 Ω µ ( x ) − Ω † Ω µ ( x ) − Ω † µ ( x ) 6 Tr µ ( x ) . 2 where T a . g 2 � 6 u a µ ( x, τ ) − w a � � 0 ∂ x,µ S G ( U ) ≃ i µ ( x, τ ) a

Perturbative matching: Wilson flow Vs. Cooling R Evolution of the Wilson flow by an infinitesimally small flow time ǫ : u a µ ( x, τ + ǫ ) ≃ u a 6 u a µ ( x, τ ) − w a � � µ ( x, τ ) − ǫ µ ( x, τ ) . a u a µ ( x, τ + ǫ ) T a where U µ ( x, τ + ǫ ) ≃ 1 1 + i � R After a cooling step: w a µ ( x, n c ) u a µ ( x, n c + 1) ≃ . 6 R Wilson flow would evolve the same as cooling if ǫ = 1 / 6. + Cooling has an additional speed up of two. ! Hence, cooling has the same effect as the Wilson flow if: τ ≃ n c 3 . Result extracted by C. Bonati and M. D’Elia, Phys. Rev. D89 (2014), 105005 [arXiv:1401.2441] R Generalization of this result for smoothing actions with rectangular terms ( b 1 ): n c τ ≃ . 3 − 15 b 1 Result extracted by C. Alexandrou, AA and K. Jansen, Phys. Rev. D92 (2015), 125014 [arXiv:1509.0425]

Numerical matching: Wilson flow Vs. Cooling Matching condition: τ ≃ n c 3 . Define function τ ( n c ) such as τ and n c change the action by the same amount. 20 0.01 τ ( n c ) Wilson Flow τ = n c / 3 Cooling 15 Average Action Density 10 0.001 τ 5 0 0.0001 0 5 10 15 20 25 30 35 40 45 50 1 10 n c τ or n c / 3