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

SLIDE 1

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.

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

SLIDE 2

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.

SLIDE 3

Overview from Lattice 2014

Continuation of Krzysztof Cichy’s talk given in Lattice 2014:

0.02 0.04 0.06 0.08 0.1 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 a χ1/4 definition ID

noSmear s=0.4 noSmear s=0 HYP1 s=0 HYP1 s=0 HYP1 s=0.75 HYP5 s=0 HYP5 s=0.5 M2=0.00003555 M2=0.0004 M2=0.001 M2=0.0015 flow time: t0 2t0 3t0 noSmear HYP10 HYP30 impr. APE10 APE30 naive APE10APE30 impr. cool10 cool30 naive cool10 cool30 impr. cool10 cool30 naive cool10cool30

index of overlap spectral flow spectral projectors field theor. GF

impr. FT noSmear
impr. FT HYP

impr./naive FT APE impr./naive FT impr. cooling impr./naive FT basic cooling

Using Nf = 2 twisted mass configuration with: β = 3.90, a ≃ 0.085fm, r0/a = 5.35(4), mπ ≃ 340 MeV, mπL = 2.5, L/a = 16

SLIDE 4

Overview from Lattice 2014

Continuation of Krzysztof Cichy’s talk given in Lattice 2014:

0.02 0.04 0.06 0.08 0.1 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 a χ1/4 definition ID

noSmear s=0.4 noSmear s=0 HYP1 s=0 HYP1 s=0 HYP1 s=0.75 HYP5 s=0 HYP5 s=0.5 M2=0.00003555 M2=0.0004 M2=0.001 M2=0.0015 flow time: t0 2t0 3t0 noSmear HYP10 HYP30 impr. APE10 APE30 naive APE10APE30 impr. cool10 cool30 naive cool10 cool30 impr. cool10 cool30 naive cool10cool30

index of overlap spectral flow spectral projectors field theor. GF

impr. FT noSmear
impr. FT HYP

impr./naive FT APE impr./naive FT impr. cooling impr./naive FT basic cooling

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻ Using Nf = 2 twisted mass configuration with:

β = 3.90, a ≃ 0.085fm, r0/a = 5.35(4), mπ ≃ 340 MeV, mπL = 2.5, L/a = 16

SLIDE 5

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

f Spectral projectors with different cutoffs M2.

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 αAPE = 0.4, 0.5, 0.6.

M. Albanese et al. [APE Collaboration], Phys. Lett. B 192 (1987) 163.

g Stout smearing with ρst = 0.01, 0.05, 0.1.

C. Morningstar and M. J. Peardon, Phys. Rev. D69 (2004) 054501

g HYP smearing with αHYP1 = 0.75, αHYP2 = 0.6 αHYP3 = 0.3.

A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (2001) 034504

SLIDE 6

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

f Spectral projectors with different cutoffs M2.

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 αAPE = 0.4, 0.5, 0.6.

M. Albanese et al. [APE Collaboration], Phys. Lett. B 192 (1987) 163.

g Stout smearing with ρst = 0.01, 0.05, 0.1.

C. Morningstar and M. J. Peardon, Phys. Rev. D69 (2004) 054501

g HYP smearing with αHYP1 = 0.75, αHYP2 = 0.6 αHYP3 = 0.3.

A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (2001) 034504

SLIDE 7

Field Theoretic Definition of the Topological Charge

g Topological charge can be defined as: Q =

d4x q(x) ,

with q(x) = 1 32π2 ǫµνρσTr {FµνFρσ} . g Discretizations of q(x) on the lattice:

R Plaquette

qplaq

L

(x) = 1 32π2 ǫµνρσTr

Cplaq

µν

Cplaq

ρσ

,

with Cplaq

µν

(x) = Im

.

R Clover

qclov

L

(x) = 1 32π2 ǫµνρσTr

Cclov

µν Cclov ρσ

,

with Cclov

µν (x) = 1

4Im

.

R Improved

qimp

L

(x) = b0qclov

L

(x) + b1qrect

L

(x) , with Crect

µν (x) = 1

8 Im      +      . g Smoothing...

SLIDE 8

Field Theoretic Definition of the Topological Charge

g Topological charge can be defined as: Q =

d4x q(x) ,

with q(x) = 1 32π2 ǫµνρσTr {FµνFρσ} . g Discretizations of q(x) on the lattice:

R Plaquette

qplaq

L

(x) = 1 32π2 ǫµνρσTr

Cplaq

µν

Cplaq

ρσ

,

with Cplaq

µν

(x) = Im

.

U Clover

qclov

L

(x) = 1 32π2 ǫµνρσTr

Cclov

µν Cclov ρσ

,

with Cclov

µν (x) = 1

4Im

.

R Improved

qimp

L

(x) = b0qclov

L

(x) + b1qrect

L

(x) , with Crect

µν (x) = 1

8 Im      +      . g Smoothing...

SLIDE 9

Example: The Wilson flow Vs. Cooling

Gradient Flow

R Solution of the evolution equations:

˙ Vµ (x, τ) = −g2

0 [∂x,µSG(V (τ))] Vµ (x, τ)

Vµ (x, 0) = Uµ (x) ,

R With link derivative defined as:

∂x,µSG(U) = i

a

T a d ds SG

eisY aU
s=0

≡ i

a

T a∂(a)

x,µSG(U) ,

R Total gradient flow time: τ R Reference flow time t0 such that t2E(t)|t=t0 = 0.3 with t = a2τ and

E(t) = − 1

2V

x Tr {Fµν(x, t)Fµν(x, t)}

Cooling

R Cooling Uµ(x) ∈ SU(N): Uold

µ

(x) → Unew

µ

(x) with

P (U) ∝ e(limβ→∞ β 1

N ReTrXµ†Uµ).

R Choose a Unew

µ

(x) that maximizes:

ReTr{U new

µ

(x)X †

µ(x)}.

R One full cooling iteration nc = 1

SLIDE 10

Perturbative expansion of links

R A link variable which has been smoothed can been written as:

Uµ(x, jsm) ≃ 1 1 + i

a

ua

µ(x, jsm)T a .

R Simple staples are written as:

per space-time slice, thus. Xµ(x, jsm) ≃ 6 · 1 1 + i

a

wa

µ(x, jsm)T a .

R For the Wilson flow with Ωµ(x) = Uµ(x)X†

µ(x)

g2

0∂x,µSG(U)(x) = 1

2

Ωµ(x) − Ω†

µ(x)

− 1

6 Tr

Ωµ(x) − Ω†

µ(x)

.

where g2

0∂x,µSG(U) ≃ i

a
6ua

µ(x, τ) − wa µ(x, τ)

T a .

SLIDE 11

Perturbative expansion of links

R A link variable which has been smoothed can been written as:

Uµ(x, jsm) ≃ 1 1 + i

a

ua

µ(x, jsm)T a .

R Simple staples are written as:

per space-time slice, thus. Xµ(x, jsm) ≃ 6 · 1 1 + i

a

wa

µ(x, jsm)T a .

R For the Wilson flow with Ωµ(x) = Uµ(x)X†

µ(x)

g2

0∂x,µSG(U)(x) = 1

2

Ωµ(x) − Ω†

µ(x)

− 1

6 Tr

Ωµ(x) − Ω†

µ(x)

.

where g2

0∂x,µSG(U) ≃ i

a
6ua

µ(x, τ) − wa µ(x, τ)

T a .

SLIDE 12

Perturbative matching: Wilson flow Vs. Cooling

R Evolution of the Wilson flow by an infinitesimally small flow time ǫ:

ua

µ(x, τ + ǫ) ≃ ua µ(x, τ) − ǫ

6ua

µ(x, τ) − wa µ(x, τ)

.

where Uµ(x, τ + ǫ) ≃ 1 1 + i

a ua µ(x, τ + ǫ)T a

R After a cooling step:

ua

µ(x, nc + 1) ≃

wa

µ(x, nc)

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 ≃ nc 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 (b1):

τ ≃ nc 3 − 15b1 .

Result extracted by C. Alexandrou, AA and K. Jansen, Phys. Rev. D92 (2015), 125014 [arXiv:1509.0425]

SLIDE 13

Numerical matching: Wilson flow Vs. Cooling

Matching condition: τ ≃ nc

3 .

Define function τ(nc) such as τ and nc change the action by the same amount.

τ = nc/3 τ(nc) nc τ 50 45 40 35 30 25 20 15 10 5 20 15 10 5 Cooling Wilson Flow τ or nc/3 Average Action Density 10 1 0.01 0.001 0.0001

SLIDE 14

Perturbative matching: Wilson flow Vs. APE

R According to the APE operation:

U(nAPE+1)

µ

(x) = ProjSU(3)

(1 − αAPE) U(nAPE)

µ

(x) + αAPE 6 X(nAPE)

µ

(x)

.

R Evolution of the Wilson flow by an infinitesimally small flow time ǫ is expressed as:

ua

µ(x, τ + ǫ) ≃ ua µ(x, τ) − ǫ

6ua

µ(x, τ) − wa µ(x, τ)

.

R Evolution of the APE smearing with parameter αAPE is expressed as:

ua

µ(x, nAPE + 1) ≃ ua µ(x, nAPE) − αAPE

6

6ua

µ(x, nAPE) − wa µ(x, nAPE)

.

U Hence, APE has the same effect as the Wilson flow if:

τ ≃ αAPE 6 nAPE .

SLIDE 15

Perturbative matching: Wilson flow Vs. APE

R According to the APE operation:

U(nAPE+1)

µ

(x) = ProjSU(3)

(1 − αAPE) U(nAPE)

µ

(x) + αAPE 6 X(nAPE)

µ

(x)

.

R Evolution of the Wilson flow by an infinitesimally small flow time ǫ is expressed as:

ua

µ(x, τ + ǫ) ≃ ua µ(x, τ) − ǫ

6ua

µ(x, τ) − wa µ(x, τ)

.

R Evolution of the APE smearing with parameter αAPE is expressed as:

ua

µ(x, nAPE + 1) ≃ ua µ(x, nAPE) − αAPE

6

6ua

µ(x, nAPE) − wa µ(x, nAPE)

.

U Hence, APE has the same effect as the Wilson flow if:

τ ≃ αAPE 6 nAPE .

SLIDE 16

Perturbative matching: Wilson flow Vs. APE

R According to the APE operation:

U(nAPE+1)

µ

(x) = ProjSU(3)

(1 − αAPE) U(nAPE)

µ

(x) + αAPE 6 X(nAPE)

µ

(x)

.

R Evolution of the Wilson flow by an infinitesimally small flow time ǫ is expressed as:

ua

µ(x, τ + ǫ) ≃ ua µ(x, τ) − ǫ

6ua

µ(x, τ) − wa µ(x, τ)

.

R Evolution of the APE smearing with parameter αAPE is expressed as:

ua

µ(x, nAPE + 1) ≃ ua µ(x, nAPE) − αAPE

6

6ua

µ(x, nAPE) − wa µ(x, nAPE)

.

U Hence, APE has the same effect as the Wilson flow if:

τ ≃ αAPE 6 nAPE .

SLIDE 17

Numerical matching: Wilson flow Vs. APE

Matching condition: τ ≃ αAPE

6 nAPE . Define function τ(αAPE, nAPE) such as τ and nAPE changes action by the same amount

0.4 6 nAPE 0.5 6 nAPE 0.6 6 nAPE

τ(0.4, nAPE) τ(0.5, nAPE) τ(0.6, nAPE)

nAPE τ 250 200 150 100 50 24 22 20 18 16 14 12 10 8 6 4 2

APE with αAPE = 0.6 APE with αAPE = 0.5 APE with αAPE = 0.4 Wilson Flow

τ or αAPE

6

× nAPE Average Action Density 10 1 0.01 0.001 0.0001

SLIDE 18

Perturbative matching: Wilson flow Vs. stout

R According to the stout smearing operation:

U(nst+1)

µ

(x) = exp

iQnst

µ

(x)

U(nst)

µ

(x) . with Qµ(x) = i 2

Ξ†

µ(x) − Ξµ(x)

− i

6 Tr

Ξ†

µ(x) − Ξµ(x)

,

with Ξµ(x) = ρstXµ(x)U†

µ(x)

R Evolution of the Wilson flow by an infinitesimally small flow time ǫ is expressed as:

ua

µ(x, τ + ǫ) ≃ ua µ(x, τ) − ǫ

6ua

µ(x, τ) − wa µ(x, τ)

.

R Evolution of the stout smearing with parameter ρst is expressed as:

ua

µ(x, nst + 1) ≃ ua µ(x, nst) − ρst

6ua

µ(x, nst) − wa µ(x, nst)

.

U Hence, stout smearing has the same effect as the Wilson flow if

τ ≃ ρstnst .

SLIDE 19

Perturbative matching: Wilson flow Vs. stout

R According to the stout smearing operation:

U(nst+1)

µ

(x) = exp

iQnst

µ

(x)

U(nst)

µ

(x) . with Qµ(x) = i 2

Ξ†

µ(x) − Ξµ(x)

− i

6 Tr

Ξ†

µ(x) − Ξµ(x)

,

with Ξµ(x) = ρstXµ(x)U†

µ(x)

R Evolution of the Wilson flow by an infinitesimally small flow time ǫ is expressed as:

ua

µ(x, τ + ǫ) ≃ ua µ(x, τ) − ǫ

6ua

µ(x, τ) − wa µ(x, τ)

.

R Evolution of the stout smearing with parameter ρst is expressed as:

ua

µ(x, nst + 1) ≃ ua µ(x, nst) − ρst

6ua

µ(x, nst) − wa µ(x, nst)

.

U Hence, stout smearing has the same effect as the Wilson flow if

τ ≃ ρstnst .

SLIDE 20

Perturbative matching: Wilson flow Vs. stout

R According to the stout smearing operation:

U(nst+1)

µ

(x) = exp

iQnst

µ

(x)

U(nst)

µ

(x) . with Qµ(x) = i 2

Ξ†

µ(x) − Ξµ(x)

− i

6 Tr

Ξ†

µ(x) − Ξµ(x)

,

with Ξµ(x) = ρstXµ(x)U†

µ(x)

R Evolution of the Wilson flow by an infinitesimally small flow time ǫ is expressed as:

ua

µ(x, τ + ǫ) ≃ ua µ(x, τ) − ǫ

6ua

µ(x, τ) − wa µ(x, τ)

.

R Evolution of the stout smearing with parameter ρst is expressed as:

ua

µ(x, nst + 1) ≃ ua µ(x, nst) − ρst

6ua

µ(x, nst) − wa µ(x, nst)

.

U Hence, stout smearing has the same effect as the Wilson flow if

τ ≃ ρstnst .

SLIDE 21

Numerical matching: Wilson flow Vs. stout

Matching condition: τ ≃ ρstnst . Define function τ(ρst, nst) such as τ and nst changes action by the same amount

0.01 × nst 0.05 × nst 0.1 × nst τ(0.01, nst) τ(0.05, nst) τ(0.1, nst)

nst τ 200 150 100 50 20 18 16 14 12 10 8 6 4 2

Stout with ρ = 0.01 Stout with ρ = 0.05 Stout with ρ = 0.1 Wilson Flow

τ or ρst × nst Average Action Density 10 1 0.01 0.001 0.0001

SLIDE 22

Numerical matching: Wilson flow Vs. HYP

R We considered HYP smearing with parameters:

αHYP1 = 0.75, αHYP2 = 0.6 αHYP3 = 0.3

D HYP staples not the same as Xµ(x) (A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (2001) 034504).

a) b)

R Define function τHYP(nHYP) and fit using the ansatz:

τHYP(nHYP) = A nHYP + B n2

HYP + C n3 HYP .

with A = 0.25447(32), B = −0.001312(90), C = 1.217(91) × 10−5

U Hence, HYP smearing has the same effect as the Wilson flow if

τ ≃ τHYP(nHYP) .

SLIDE 23

Numerical matching: Wilson flow Vs. HYP

R We considered HYP smearing with parameters:

αHYP1 = 0.75, αHYP2 = 0.6 αHYP3 = 0.3

D HYP staples not the same as Xµ(x) (A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (2001) 034504).

a) b)

R Define function τHYP(nHYP) and fit using the ansatz:

τHYP(nHYP) = A nHYP + B n2

HYP + C n3 HYP .

with A = 0.25447(32), B = −0.001312(90), C = 1.217(91) × 10−5

U Hence, HYP smearing has the same effect as the Wilson flow if

τ ≃ τHYP(nHYP) .

SLIDE 24

Numerical matching: Wilson flow Vs. HYP

Numerical matching condition: τHYP(nHYP) = A nHYP + B n2

HYP + C n3 HYP .

Define function τ(nHYP) such as τ and nHYP changes action by the same amount

τHYP(nHYP) 0.245 × nHYP τ(nHYP) nHYP τ 50 45 40 35 30 25 20 15 10 5 14 12 10 8 6 4 2 HYP smearing Wilson Flow τ or τHYP(nHYP) Average Action Density 10 1 0.01 0.001 0.0001

SLIDE 25

Topological Susceptibility - The Wilson flow

The Wilson flow time t0 ≃ 2.5a2

Gradient flow τ r0χ1/4 10 8 6 4 2 0.5 0.45 0.4 0.35 0.3

χ = Q2

V

SLIDE 26

Topological Susceptibility - Cooling

The Wilson flow time t0 ≃ 2.5a2 ≡ nc = 7.5 cooling steps

Cooling Gradient flow τ or nc/3 r0χ1/4 10 8 6 4 2 0.5 0.45 0.4 0.35 0.3

χ = Q2

V

SLIDE 27

Topological Susceptibility - APE

The Wilson flow time t0 ≃ 2.5a2 ≡ nAPE = 37.5 APE smearing steps for αAPE = 0.4

APE smearing αAPE = 0.4 Gradient flow τ or αAPE × nAPE/6 r0χ1/4 10 8 6 4 2 0.5 0.45 0.4 0.35 0.3

χ = Q2

V

SLIDE 28

Topological Susceptibility - APE

The Wilson flow time t0 ≃ 2.5a2 ≡ nAPE = 30 APE smearing steps for αAPE = 0.5

APE smearing αAPE = 0.5 APE smearing αAPE = 0.4 Gradient flow τ or αAPE × nAPE/6 r0χ1/4 10 8 6 4 2 0.5 0.45 0.4 0.35 0.3

χ = Q2

V

SLIDE 29

Topological Susceptibility - APE

The Wilson flow time t0 ≃ 2.5a2 ≡ nAPE = 25 APE smearing steps for αAPE = 0.6

APE smearing αAPE = 0.6 APE smearing αAPE = 0.5 APE smearing αAPE = 0.4 Gradient flow τ or αAPE × nAPE/6 r0χ1/4 10 8 6 4 2 0.5 0.45 0.4 0.35 0.3

χ = Q2

V

SLIDE 30

Topological Susceptibility - stout

The Wilson flow time t0 ≃ 2.5a2 ≡ nst = 250 stout smearing steps for ρst = 0.01

stout smearing ρst = 0.01 Gradient flow τ or ρst × nst r0χ1/4 10 8 6 4 2 0.5 0.45 0.4 0.35 0.3

χ = Q2

V

SLIDE 31

Topological Susceptibility - stout

The Wilson flow time t0 ≃ 2.5a2 ≡ nst = 50 stout smearing steps for ρst = 0.05

stout smearing ρst = 0.05 stout smearing ρst = 0.01 Gradient flow τ or ρst × nst r0χ1/4 10 8 6 4 2 0.5 0.45 0.4 0.35 0.3

χ = Q2

V

SLIDE 32

Topological Susceptibility - stout

The Wilson flow time t0 ≃ 2.5a2 ≡ nst = 25 stout smearing steps for ρst = 0.1

stout smearing ρst = 0.1 stout smearing ρst = 0.05 stout smearing ρst = 0.01 Gradient flow τ or ρst × nst r0χ1/4 10 8 6 4 2 0.5 0.45 0.4 0.35 0.3

χ = Q2

V

SLIDE 33

Topological Susceptibility - HYP

The Wilson flow time t0 ≃ 2.5a2 ≡ nst = 10 HYP smearing steps

HYP smearing Gradient flow τ or τHYP r0χ1/4 10 8 6 4 2 0.5 0.45 0.4 0.35 0.3

χ = Q2

V

SLIDE 34

Correlation between different smoothers

Let us have a look at the correlation coefficient cQ1,Q2 =

Q1 − Q1

Q2 − Q2

Q1 − Q1

2 Q2 − Q2 2 . WF, t0 cool, t0 APE, t0 stout, t0 HYP, t0 WF, t0 1.00(0) 0.97(0) 1.00(0) 1.00(0) 0.97(0) cool, t0 0.97(0) 1.00(0) 0.97(0) 0.97(0) 0.94(0) APE, t0 1.00(0) 0.97(0) 1.00(0) 1.00(0) 0.97(0) stout, t0 1.00(0) 0.97(0) 1.00(0) 1.00(0) 0.97(0) HYP, t0 0.97(0) 0.94(0) 0.97(0) 0.97(0) 1.00(0) Topological charges are highly correlated! In the continuum all numbers become 1.00

SLIDE 35

Fixing the smoothing scale

U One can fix a physical flow time:

λS ≃ √ 8t .

R

Similarly for cooling: λS ≃ a

8nc

3 .

R

For the APE smearing: λS ≃ a

4αAPEnAPE

3 .

R

For the stout smearing λS ≃ a

8ρstnst .

R Similar procedure can be applied to t0?

SLIDE 36

General comparison: Topological Susceptibility

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

Comparison of results for the topological susceptibility.

0.02 0.04 0.06 0.08 0.1 1 2 3 4 5 6 7 8 9 10 11 12 16 18 22 24 25 27 34 35 28 30 40 41 44 45 a χ1/4 definition

noSmear s=0.4 noSmear s=0 HYP1 s=0 HYP1 s=0 HYP1 s=0.75 HYP5 s=0 HYP5 s=0.5

M2=0.00003555 M2=0.0004 M2=0.001 M2=0.0015

t0 3t0 Wplaq t0 3t0 tlSym Iwa t0 3t0 noSmear stout t0 3t0 cool Wplaq t0 3t0 APE t0 3t0 HYP t0 3t0

index of overlap spectral flow spectral projectors cFT noSmear cFT GF cFT stout cFT cooling cFT APE cFT HYP

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

Using Nf = 2 twisted mass configuration with: β = 3.90, a ≃ 0.085fm, r0/a = 5.35(4), mπ ≃ 340 MeV, mπL = 2.5, L/a = 16

SLIDE 37

General comparison: Correlation Coefficient

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

Comparison of the correlation coefficient between fermionic and gluonic definitions.

index nonSmear s=0.4 | 1 index HYP1 s=0 | 2 SF HYP1 s=0.0 | 3 SF HYP5 s=0.0 | 4

spec. proj. M2=0.0004 | 5
spec. proj. M2=0.0010 | 6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

definition 2 definition 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SLIDE 38

Continuum limit

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

Correlation for a fermionic and gluonic definitions as we approach the continuum limit.

index with HYP1, s=0 vs. the Wilson flow at t0 a/r0 correlation 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.8

SLIDE 39

Conclusions

R Topological susceptibilities are in the same ballpark: aχ1/4 ∈ [0.08, 0.09]. R Correlation coefficient increases towards to 1 as a → 0. R Different definitions influenced by different lattice artifacts. R Most correlation coefficients are above 80 %. R Cooling, APE smearing, stout smearing are numerically equivalent if matched:

τ ≃ nc 3 , τ ≃ αAPE nAPE 6 τ ≃ ρstnst . ∼ 120 × faster, ∼ 20 × faster ∼ 30 × faster .

SLIDE 40

Conclusions

R Topological susceptibilities are in the same ballpark: aχ1/4 ∈ [0.08, 0.09]. R Correlation coefficient increases towards to 1 as a → 0. R Different definitions influenced by different lattice artifacts. R Most correlation coefficients are above 80 %. R Cooling, APE smearing, stout smearing are numerically equivalent if matched:

τ ≃ nc 3 , τ ≃ αAPE nAPE 6 τ ≃ ρstnst . ∼ 120 × faster, ∼ 20 × faster ∼ 30 × faster .

SLIDE 41

Conclusions

THANK YOU!

SLIDE 42

Appendix: General comparison: Correlation Coefficient

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

Comparison of the correlation coefficient between fermionic and gluonic definitions.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 1.00(0) 0.96(0) 0.95(0) 0.92(1) 0.58(4) 0.60(3) 0.18(6) 0.86(1) 0.90(1) 0.93(0) 0.86(1) 0.89(1) 0.91(0) 0.86(1) 0.86(1) 0.91(1) 2 0.96(0) 1.00(0) 0.99(0) 0.93(0) 0.54(4) 0.62(3) 0.17(4) 0.88(1) 0.92(0) 0.95(0) 0.87(1) 0.91(1) 0.94(0) 0.88(1) 0.88(1) 0.92(0) 3 0.95(0) 0.99(0) 1.00(0) 0.93(0) 0.54(4) 0.62(3) 0.17(4) 0.88(1) 0.91(0) 0.95(0) 0.86(1) 0.90(0) 0.93(0) 0.88(1) 0.88(1) 0.92(0) 4 0.92(1) 0.93(0) 0.93(0) 1.00(0) 0.56(4) 0.61(3) 0.15(4) 0.92(0) 0.96(0) 0.91(0) 0.90(1) 0.93(0) 0.91(0) 0.92(0) 0.92(0) 0.97(0) 5 0.58(4) 0.54(4) 0.54(4) 0.56(4) 1.00(0) 0.62(4) 0.10(3) 0.66(3) 0.63(3) 0.56(4) 0.66(3) 0.62(3) 0.56(4) 0.65(3) 0.65(3) 0.62(3) 6 0.60(3) 0.62(3) 0.62(3) 0.61(3) 0.62(4) 1.00(0) 0.09(4) 0.67(3) 0.65(3) 0.60(4) 0.68(3) 0.66(3) 0.61(4) 0.66(3) 0.66(3) 0.65(3) 7 0.18(6) 0.17(4) 0.17(4) 0.15(4) 0.10(3) 0.09(4) 1.00(0) 0.16(4) 0.18(4) 0.17(4) 0.15(4) 0.18(4) 0.18(4) 0.16(4) 0.16(4) 0.17(4) 8 0.86(1) 0.88(1) 0.88(1) 0.92(0) 0.66(3) 0.67(3) 0.16(4) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.96(0) 0.88(1) 1.00(0) 1.00(0) 0.97(0) 9 0.90(1) 0.92(0) 0.91(0) 0.96(0) 0.63(3) 0.65(3) 0.18(4) 0.97(0) 1.00(0) 0.92(0) 0.94(0) 0.96(0) 0.92(0) 0.97(0) 0.97(0) 0.99(0) 10 0.93(0) 0.95(0) 0.95(0) 0.91(0) 0.56(4) 0.60(4) 0.17(4) 0.88(1) 0.92(0) 1.00(0) 0.86(1) 0.90(1) 0.95(0) 0.88(1) 0.88(1) 0.91(0) 11 0.86(1) 0.87(1) 0.86(1) 0.90(1) 0.66(3) 0.68(3) 0.15(4) 0.97(0) 0.94(0) 0.86(1) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.97(0) 0.94(0) 12 0.89(1) 0.91(1) 0.90(0) 0.93(0) 0.62(3) 0.66(3) 0.18(4) 0.96(0) 0.96(0) 0.90(1) 0.97(0) 1.00(0) 0.92(0) 0.96(0) 0.96(0) 0.96(0) 13 0.91(0) 0.94(0) 0.93(0) 0.91(0) 0.56(4) 0.61(4) 0.18(4) 0.88(1) 0.92(0) 0.95(0) 0.88(1) 0.92(0) 1.00(0) 0.88(1) 0.88(1) 0.91(0) 14 0.86(1) 0.88(1) 0.88(1) 0.92(0) 0.65(3) 0.66(3) 0.16(4) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.96(0) 0.88(1) 1.00(0) 1.00(0) 0.97(0) 15 0.86(1) 0.88(1) 0.88(1) 0.92(0) 0.65(3) 0.66(3) 0.16(4) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.96(0) 0.88(1) 1.00(0) 1.00(0) 0.97(0) 16 0.91(1) 0.92(0) 0.92(0) 0.97(0) 0.62(3) 0.65(3) 0.17(4) 0.97(0) 0.99(0) 0.91(0) 0.94(0) 0.96(0) 0.91(0) 0.97(0) 0.97(0) 1.00(0)

SLIDE 43

Appendix: General comparison: Correlation Coefficient

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

Comparison of the correlation coefficient between fermionic and gluonic definitions.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 1.00(0) 0.96(0) 0.95(0) 0.92(1) 0.58(4) 0.60(3) 0.18(6) 0.86(1) 0.90(1) 0.93(0) 0.86(1) 0.89(1) 0.91(0) 0.86(1) 0.86(1) 0.91(1) 2 0.96(0) 1.00(0) 0.99(0) 0.93(0) 0.54(4) 0.62(3) 0.17(4) 0.88(1) 0.92(0) 0.95(0) 0.87(1) 0.91(1) 0.94(0) 0.88(1) 0.88(1) 0.92(0) 3 0.95(0) 0.99(0) 1.00(0) 0.93(0) 0.54(4) 0.62(3) 0.17(4) 0.88(1) 0.91(0) 0.95(0) 0.86(1) 0.90(0) 0.93(0) 0.88(1) 0.88(1) 0.92(0) 4 0.92(1) 0.93(0) 0.93(0) 1.00(0) 0.56(4) 0.61(3) 0.15(4) 0.92(0) 0.96(0) 0.91(0) 0.90(1) 0.93(0) 0.91(0) 0.92(0) 0.92(0) 0.97(0) 5 0.58(4) 0.54(4) 0.54(4) 0.56(4) 1.00(0) 0.62(4) 0.10(3) 0.66(3) 0.63(3) 0.56(4) 0.66(3) 0.62(3) 0.56(4) 0.65(3) 0.65(3) 0.62(3) 6 0.60(3) 0.62(3) 0.62(3) 0.61(3) 0.62(4) 1.00(0) 0.09(4) 0.67(3) 0.65(3) 0.60(4) 0.68(3) 0.66(3) 0.61(4) 0.66(3) 0.66(3) 0.65(3) 7 0.18(6) 0.17(4) 0.17(4) 0.15(4) 0.10(3) 0.09(4) 1.00(0) 0.16(4) 0.18(4) 0.17(4) 0.15(4) 0.18(4) 0.18(4) 0.16(4) 0.16(4) 0.17(4) 8 0.86(1) 0.88(1) 0.88(1) 0.92(0) 0.66(3) 0.67(3) 0.16(4) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.96(0) 0.88(1) 1.00(0) 1.00(0) 0.97(0) 9 0.90(1) 0.92(0) 0.91(0) 0.96(0) 0.63(3) 0.65(3) 0.18(4) 0.97(0) 1.00(0) 0.92(0) 0.94(0) 0.96(0) 0.92(0) 0.97(0) 0.97(0) 0.99(0) 10 0.93(0) 0.95(0) 0.95(0) 0.91(0) 0.56(4) 0.60(4) 0.17(4) 0.88(1) 0.92(0) 1.00(0) 0.86(1) 0.90(1) 0.95(0) 0.88(1) 0.88(1) 0.91(0) 11 0.86(1) 0.87(1) 0.86(1) 0.90(1) 0.66(3) 0.68(3) 0.15(4) 0.97(0) 0.94(0) 0.86(1) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.97(0) 0.94(0) 12 0.89(1) 0.91(1) 0.90(0) 0.93(0) 0.62(3) 0.66(3) 0.18(4) 0.96(0) 0.96(0) 0.90(1) 0.97(0) 1.00(0) 0.92(0) 0.96(0) 0.96(0) 0.96(0) 13 0.91(0) 0.94(0) 0.93(0) 0.91(0) 0.56(4) 0.61(4) 0.18(4) 0.88(1) 0.92(0) 0.95(0) 0.88(1) 0.92(0) 1.00(0) 0.88(1) 0.88(1) 0.91(0) 14 0.86(1) 0.88(1) 0.88(1) 0.92(0) 0.65(3) 0.66(3) 0.16(4) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.96(0) 0.88(1) 1.00(0) 1.00(0) 0.97(0) 15 0.86(1) 0.88(1) 0.88(1) 0.92(0) 0.65(3) 0.66(3) 0.16(4) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.96(0) 0.88(1) 1.00(0) 1.00(0) 0.97(0) 16 0.91(1) 0.92(0) 0.92(0) 0.97(0) 0.62(3) 0.65(3) 0.17(4) 0.97(0) 0.99(0) 0.91(0) 0.94(0) 0.96(0) 0.91(0) 0.97(0) 0.97(0) 1.00(0)

SLIDE 44

Appendix: General comparison: Correlation Coefficient

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

Comparison of the correlation coefficient between fermionic and gluonic definitions.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 1.00(0) 0.96(0) 0.95(0) 0.92(1) 0.58(4) 0.60(3) 0.18(6) 0.86(1) 0.90(1) 0.93(0) 0.86(1) 0.89(1) 0.91(0) 0.86(1) 0.86(1) 0.91(1) 2 0.96(0) 1.00(0) 0.99(0) 0.93(0) 0.54(4) 0.62(3) 0.17(4) 0.88(1) 0.92(0) 0.95(0) 0.87(1) 0.91(1) 0.94(0) 0.88(1) 0.88(1) 0.92(0) 3 0.95(0) 0.99(0) 1.00(0) 0.93(0) 0.54(4) 0.62(3) 0.17(4) 0.88(1) 0.91(0) 0.95(0) 0.86(1) 0.90(0) 0.93(0) 0.88(1) 0.88(1) 0.92(0) 4 0.92(1) 0.93(0) 0.93(0) 1.00(0) 0.56(4) 0.61(3) 0.15(4) 0.92(0) 0.96(0) 0.91(0) 0.90(1) 0.93(0) 0.91(0) 0.92(0) 0.92(0) 0.97(0) 5 0.58(4) 0.54(4) 0.54(4) 0.56(4) 1.00(0) 0.62(4) 0.10(3) 0.66(3) 0.63(3) 0.56(4) 0.66(3) 0.62(3) 0.56(4) 0.65(3) 0.65(3) 0.62(3) 6 0.60(3) 0.62(3) 0.62(3) 0.61(3) 0.62(4) 1.00(0) 0.09(4) 0.67(3) 0.65(3) 0.60(4) 0.68(3) 0.66(3) 0.61(4) 0.66(3) 0.66(3) 0.65(3) 7 0.18(6) 0.17(4) 0.17(4) 0.15(4) 0.10(3) 0.09(4) 1.00(0) 0.16(4) 0.18(4) 0.17(4) 0.15(4) 0.18(4) 0.18(4) 0.16(4) 0.16(4) 0.17(4) 8 0.86(1) 0.88(1) 0.88(1) 0.92(0) 0.66(3) 0.67(3) 0.16(4) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.96(0) 0.88(1) 1.00(0) 1.00(0) 0.97(0) 9 0.90(1) 0.92(0) 0.91(0) 0.96(0) 0.63(3) 0.65(3) 0.18(4) 0.97(0) 1.00(0) 0.92(0) 0.94(0) 0.96(0) 0.92(0) 0.97(0) 0.97(0) 0.99(0) 10 0.93(0) 0.95(0) 0.95(0) 0.91(0) 0.56(4) 0.60(4) 0.17(4) 0.88(1) 0.92(0) 1.00(0) 0.86(1) 0.90(1) 0.95(0) 0.88(1) 0.88(1) 0.91(0) 11 0.86(1) 0.87(1) 0.86(1) 0.90(1) 0.66(3) 0.68(3) 0.15(4) 0.97(0) 0.94(0) 0.86(1) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.97(0) 0.94(0) 12 0.89(1) 0.91(1) 0.90(0) 0.93(0) 0.62(3) 0.66(3) 0.18(4) 0.96(0) 0.96(0) 0.90(1) 0.97(0) 1.00(0) 0.92(0) 0.96(0) 0.96(0) 0.96(0) 13 0.91(0) 0.94(0) 0.93(0) 0.91(0) 0.56(4) 0.61(4) 0.18(4) 0.88(1) 0.92(0) 0.95(0) 0.88(1) 0.92(0) 1.00(0) 0.88(1) 0.88(1) 0.91(0) 14 0.86(1) 0.88(1) 0.88(1) 0.92(0) 0.65(3) 0.66(3) 0.16(4) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.96(0) 0.88(1) 1.00(0) 1.00(0) 0.97(0) 15 0.86(1) 0.88(1) 0.88(1) 0.92(0) 0.65(3) 0.66(3) 0.16(4) 1.00(0) 0.97(0) 0.88(1) 0.97(0) 0.96(0) 0.88(1) 1.00(0) 1.00(0) 0.97(0) 16 0.91(1) 0.92(0) 0.92(0) 0.97(0) 0.62(3) 0.65(3) 0.17(4) 0.97(0) 0.99(0) 0.91(0) 0.94(0) 0.96(0) 0.91(0) 0.97(0) 0.97(0) 1.00(0)

SLIDE 45

Appendix: General comparison: Correlation Coefficient

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

Comparison of correlation coefficient between cooling and the gradient flow.

β = 2.10 β = 1.95 β = 1.90

nc and 3 × τ cQ1,Q2

60 50 40 30 20 10 1 0.975 0.95 0.925 0.9 0.875 0.85 0.825 0.8 0.775 0.75 0.725 0.7 0.675 0.65 0.625 0.6 0.575 0.55 0.525 0.5

Wilson

β = 2.10 β = 1.95 β = 1.90

nc and 4.25 × τ

60 50 40 30 20 10

Symanzik tree-level

β = 2.10 β = 1.95 β = 1.90

nc and 7.965 × τ

60 50 40 30 20 10

Iwasaki

SLIDE 46

Appendix: General comparison: Correlation Coefficient

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

Comparison of correlation coefficient between cooling and the gradient flow. 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.9 0.89 0.88 0.87

3 × τ

60 50 40 30 20 10

nc

60 50 40 30 20 10

Wilson

0.98

0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.9

4.25 × τ

60 50 40 30 20 10

nc

60 50 40 30 20 10

Symanzik tree-level

SLIDE 47

Appendix: General comparison: Distribution

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

Comparison of distributions between cooling and the gradient flow.

cooling gradient flow

confs

300 250 200 150 100 50

Wilson

cooling gradient flow

Symanzik tree-level

cooling gradient flow

Iwasaki

cooling (χ2

d.o.f ≃ 1.47)

gradient flow (χ2

d.o.f ≃ 1.62)

Q confs

40 20

20
40

300 250 200 150 100 50

Wilson

cooling (χ2

d.o.f ≃ 1.21)

gradient flow (χ2

d.o.f ≃ 1.75)

Q

40 20

20
40

Symanzik tree-level

cooling (χ2

d.o.f ≃ 1.27)

gradient flow (χ2

d.o.f ≃ 1.44)

Q

40 20

20
40

Iwasaki

SLIDE 48

Appendix: General comparison: τ(nc)

▲ ❆ ❚ ❚■❈ ❊ ✷ ✵ ✶ ✻

Comparison of τ(nc) for different smoothing actions. Prediction τ ≃

nc 3−15b1 . β = 2.10 β = 1.95 β = 1.90

nc τ(nc)

35 30 25 20 15 10 5 20 15 10 5

Wilson

β = 2.10 β = 1.95 β = 1.90

nc

35 30 25 20 15 10 5

Symanzik tree-level

β = 2.10 β = 1.95 β = 1.90

nc

40 35 30 25 20 15 10 5

Iwasaki

SLIDE 49

Appendix: Topological Charge: level of agreement

Why topological susceptibility has such a high level of agreement?

stout with ρst = 0.05 APE with αAPE = 0.5 HYP cooling Wilson flow τ Q 10 8 6 4 2 1.5 1 0.5

0.5
1
1.5
2
2.5
3

SLIDE 50