Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 1 / 35

Comparison of lattice definitions of the topological charge

Krzysztof CichyNIC, DESY Zeuthen, Germany

Adam Mickiewicz University, Poznań, Poland

in collaboration with:Arthur Dromard, Elena Garcia-Ramos, Karl Jansen, Konstantin Ottnad,

Carsten Urbach, Marc Wagner, Urs Wenger, Falk Zimmermann

Presentation outline

Presentationoutline

Introduction

Results

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 2 / 35

1. Introduction

• Motivation

• Short overview of employed definitions

2. Results

• Comparison at a single lattice spacing

• Increase of correlation towards the continuum limit

• Topological susceptibility

3. Conclusions

Motivation

Presentationoutline

Introduction

Motivation

Definitions

Index

Spectral flow

Spectral projectors

Fermionic disc.

Field theoretic

Lattice setup

Results

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 3 / 35

• theoretical

⋆ there are many definitions of the topological charge

⋆ which of them should one use for different purposes?

⋄ to compute the topological susceptibility

⋄ to sort configurations according to the topological charge

⋄ for weighting results with the topological charge

⋆ what are the pros and cons of different definitions?

⋆ which definitions are theoretically clean and which are“suspicious”?

• practical

⋆ quite a lot of unpublished data for the topological charge fromdifferent definitions and several projects

⋆ missing data rather easy to compute

Definitions

Presentationoutline

Introduction

Motivation

Definitions

Index

Spectral flow

Spectral projectors

Fermionic disc.

Field theoretic

Lattice setup

Results

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 4 / 35

• index of the overlap Dirac operator [K.C., E. García Ramos, K. Jansen]

• spectral flow of the Hermitian Wilson-Dirac operator [U. Wenger]

• spectral projectors [K.C., E. García Ramos, K. Jansen]

• fermionic from disconnected loops [K. Ottnad, C. Urbach,

F. Zimmermann]

• field theoretic with HYP smearing [U. Wenger, F. Zimmermann]

• field theoretic with APE smearing [A. Dromard, M. Wagner,

F. Zimmermann]

• field theoretic with cooling [A. Dromard, M. Wagner]

• field theoretic with gradient flow [U. Wenger]

Index of the overlap Dirac operator

Presentationoutline

Introduction

Motivation

Definitions

Index

Spectral flow

Spectral projectors

Fermionic disc.

Field theoretic

Lattice setup

Results

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 5 / 35

• Chirally symmetric fermionic discretizations allow exact zero modesof the Dirac operator.

• The famous Atiyah-Singer index theorem [M. Atiyah, I.M. Singer, Ann.

Math. 93, 139 (1971) 168] relates the topological charge to the numberof zero modes of the Dirac operator:

Q = n− − n+

• This is quite remarkable, because a property of gauge fields is linkedto a fermionic observable.

• uniqueness: dependence on the s parameter of the kernel

• pros: theoretically clean, integer-valued, no renormalization

• cons: cost, cost, cost...

Spectral flow of

the Hermitian Wilson-Dirac operator

Presentationoutline

Introduction

Motivation

Definitions

Index

Spectral flow

Spectral projectors

Fermionic disc.

Field theoretic

Lattice setup

Results

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 6 / 35

• closely related and actually equivalent to the index

• uniqueness: dependence on the s parameter of the kernel

• pros: theoretically clean, integer-valued, no renormalization, onecomputation leads to the whole s-dependence of the index

• cons: cost (although still cheaper than index), non-trivial to analyzedata

Spectral projectors

Presentationoutline

Introduction

Motivation

Definitions

Index

Spectral flow

Spectral projectors

Fermionic disc.

Field theoretic

Lattice setup

Results

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 7 / 35

• another fermionic definition, introduced in:[L. Giusti, M. Lüscher, 2008], [M. Lüscher, F. Palombi, 2010]

• PM – projector to the subspace of eigenmodes of D†D witheigenvalues below M2, evaluated stochastically

• Q = Tr {γ5PM} for chirally symmetric fermions

• spectral projectors are then also equivalent to the index (=stochasticway of counting the zero modes)

• for non-chirally symmetric fermions it still gives a clean definition,although chirality of modes is no longer ±1 −→ ±1 +O(a2)

• in practice, one evaluates the observable:

C =1

N

N∑k=1

(PMηk, γ5PMηk) ,• uniqueness: dependence on the M of PM• pros: theoretically clean, still rather cheap

• cons: stochastic ingredient, non-integer, ZS/ZP needed

Fermionic from disconnected loops

Presentationoutline

Introduction

Motivation

Definitions

Index

Spectral flow

Spectral projectors

Fermionic disc.

Field theoretic

Lattice setup

Results

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 8 / 35

• another fermionic definition, given by Chris Michael:

NfQ = mq∑

ψ̄γ5ψ =∑ mqγ5

D +mq,

in the limit as mq → 0.

• allows for a Q computation as a by-product of evaluation ofdisconnected loops

• uniqueness: yes

• pros: cheap if treated as a by-product

• cons: unclear to what extent it is valid, stochastic ingredient,non-integer, ZS/ZP needed

Field theoretic

Presentationoutline

Introduction

Motivation

Definitions

Index

Spectral flow

Spectral projectors

Fermionic disc.

Field theoretic

Lattice setup

Results

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 9 / 35

• a very natural definition

• in the continuum:

Q =1

32π2

∫d4x ǫµνρσtr[Fµν(x)Fρσ]

• on the lattice one has to choose some discretization

• renormalization:qR[U ] = round(Zqbare[U ]),

Z is the (non-zero) solution of:

min∑U

(Zqbare − round(Zqbare[U ]))2

• smoothing of gauge fields needed

Field theoretic

Presentationoutline

Introduction

Motivation

Definitions

Index

Spectral flow

Spectral projectors

Fermionic disc.

Field theoretic

Lattice setup

Results

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 10 / 35

• smoothing:

⋆ cooling – an iterative minimization of the lattice action,eliminates rough topological fluctuations while keeping largeinstantons unchanged and decreases renormalization effects bysmoothing out the UV noise[B. Berg, 1981], [Y. Iwasaki et al., 1983], [M. Teper, 1985],

[E. Ilgenfritz et al., 1986]

⋆ HYP/APE smearing [M. Albanese et al., 1987], [A. Hasenfratz,

F. Knechtli, 2001]

⋆ gradient flow (GF) [M. Lüscher, 2010]

• GF is very important from the point of view of validity of the fieldtheoretic approach

• uniqueness: discretization of F , level of smoothing

• pros: very cheap (but: cost of smoothing), theor. clean if GF usedfor smoothing and no renorm. then [M. Lüscher, P. Weisz, 2011]

• cons: HYP/APE or cooling a bit ad hoc (require renorm.)

Lattice setup

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 11 / 35

Ensemble β lattice aµl µR [MeV] κc L [fm] mπLb40.16 3.90 163 × 32 0.004 21 0.160856 1.4 2.5c30.20 4.05 203 × 40 0.003 19 0.157010 1.3 2.4d20.24 4.20 243 × 48 0.002 15 0.154073 1.3 2.4e17.32 4.35 323 × 64 0.00175 16 0.151740 1.5 2.4

Ensemble β lattice aµl µl,R [MeV] κc L [fm] mπLA30.32 1.90 323 × 64 0.0030 13 0.163272 2.8 4.0A40.32 1.90 323 × 64 0.0040 17 0.163270 2.8 4.5A50.32 1.90 323 × 64 0.0050 22 0.163267 2.8 5.1A60.24 1.90 243 × 48 0.0060 26 0.163265 2.1 4.2A80.24 1.90 243 × 48 0.0080 35 0.163260 2.1 4.8B25.32 1.95 323 × 64 0.0025 13 0.161240 2.5 3.4B35.32 1.95 323 × 64 0.0035 18 0.161240 2.5 4.0B55.32 1.95 323 × 64 0.0055 28 0.161236 2.5 5.0B75.32 1.95 323 × 64 0.0075 38 0.161232 2.5 5.8B85.24 1.95 243 × 48 0.0085 45 0.161231 1.9 4.7D20.48 2.10 483 × 96 0.0020 12 0.156357 2.9 3.9D30.48 2.10 483 × 96 0.0030 19 0.156355 2.9 4.7D45.32 2.10 323 × 64 0.0045 29 0.156315 1.9 3.9

MC histories – b40.16

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 12 / 35

-8

-6

-4

-2

0

2

4

6

8

10

12

2000 2500 3000 3500 4000

Q (

index n

onS

mear

s=

0.4

)

conf number

-8

-6

-4

-2

0

2

4

6

8

10

12

1500 2000 2500 3000 3500 4000

Q (

index H

YP

1 s

=0)

conf number

-10

-5

0

5

10

15

1500 2000 2500 3000 3500 4000

Q (

SF

HY

P1 s

=0.7

5)

conf number

-8

-6

-4

-2

0

2

4

6

8

10

12

1500 2000 2500 3000 3500 4000

Q (

SF

HY

P1 s

=0.0

)

conf number

-10

-8

-6

-4

-2

0

2

4

6

8

10

1500 2000 2500 3000 3500 4000

Q (

SF

HY

P5 s

=0.5

)

conf number

-8

-6

-4

-2

0

2

4

6

8

10

1500 2000 2500 3000 3500 4000

Q (

SF

HY

P5 s

=0.0

)

conf number

-3

-2

-1

0

1

2

3

4

5

1500 2000 2500 3000 3500 4000

Q (

spec. pro

j. n

onS

mear)

conf number

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

1500 2000 2500 3000 3500 4000

Q (

ferm

ionic

)

conf number

-8

-6

-4

-2

0

2

4

6

8

500 1000 1500 2000 2500 3000 3500 4000

Q (

GF

flo

w tim

e t0)

conf number

-8

-6

-4

-2

0

2

4

6

8

10

500 1000 1500 2000 2500 3000 3500 4000

Q (

GF

flo

w tim

e 2

t0)

conf number

-8

-6

-4

-2

0

2

4

6

8

10

500 1000 1500 2000 2500 3000 3500 4000

Q (

GF

flo

w tim

e 3

t0)

conf number

-8

-6

-4

-2

0

2

4

6

2000 2500 3000 3500 4000

Q (

field

theor.

nonS

mear)

conf number

-8

-6

-4

-2

0

2

4

6

8

10

1500 2000 2500 3000 3500 4000

Q (

field

theor.

HY

P10)

conf number

-8

-6

-4

-2

0

2

4

6

8

10

1500 2000 2500 3000 3500 4000

Q (

field

theor.

HY

P40)

conf number

-3

-2

-1

0

1

2

3

4

1500 2000 2500 3000 3500 4000

Q (

field

theor.

AP

E10)

conf number

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

2000 2500 3000 3500 4000

Q (

field

theor.

AP

E40)

conf number

-8

-6

-4

-2

0

2

4

6

8

10

1500 2000 2500 3000 3500 4000

Q (

field

theor.

coolin

g tol5

)

conf number

-8

-6

-4

-2

0

2

4

6

8

10

1500 2000 2500 3000 3500 4000

Q (

field

theor.

coolin

g tol1

)

conf number

Histograms – b40.16

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 13 / 35

0

5

10

15

20

25

30

35

40

45

-10 -5 0 5 10 15

num

ber

of confs

Q (index nonSmear s=0.4)

0

10

20

30

40

50

60

70

80

90

-10 -5 0 5 10 15

num

ber

of confs

Q (index HYP1 s=0)

0

10

20

30

40

50

60

70

80

-15 -10 -5 0 5 10 15

num

ber

of confs

Q (SF HYP1 s=0.75)

0

10

20

30

40

50

60

70

80

-10 -5 0 5 10 15

num

ber

of confs

Q (SF HYP1 s=0.0)

0

10

20

30

40

50

60

70

80

90

-10 -5 0 5 10

num

ber

of confs

Q (SF HYP5 s=0.5)

0

10

20

30

40

50

60

70

80

90

-10 -8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (SF HYP5 s=0.0)

0

20

40

60

80

100

120

-4 -3 -2 -1 0 1 2 3 4 5

num

ber

of confs

Q (spec. proj. nonSmear)

0

10

20

30

40

50

60

70

80

90

-3 -2 -1 0 1 2 3

num

ber

of confs

Q (fermionic)

0

10

20

30

40

50

60

70

80

90

-8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (GF flow time t0)

0

10

20

30

40

50

60

70

80

-8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (GF flow time 2t0)

0

10

20

30

40

50

60

70

80

90

-8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (GF flow time 3t0)

0

5

10

15

20

25

30

-8 -6 -4 -2 0 2 4 6

num

ber

of confs

Q (field theor. nonSmear)

0

5

10

15

20

25

-8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (field theor. HYP10)

0

5

10

15

20

25

30

35

40

-8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (field theor. HYP40)

0

5

10

15

20

25

30

35

-3 -2 -1 0 1 2 3 4

num

ber

of confs

Q (field theor. APE10)

0

5

10

15

20

25

30

35

40

45

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

num

ber

of confs

Q (field theor. APE40)

0

5

10

15

20

25

-8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (field theor. cooling tol5)

0

5

10

15

20

25

30

35

40

-8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (field theor. cooling tol1)

Histograms – b40.16

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 14 / 35

0

5

10

15

20

25

-8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (field theor. HYP10)

0

5

10

15

20

25

30

35

40

-8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (field theor. HYP40)

Field theoretic definition

10 HYP iterations 40 HYP iterations

Histograms – b40.16

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 15 / 35

0

5

10

15

20

25

-8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (field theor. cooling tol5)

0

5

10

15

20

25

30

35

40

-8 -6 -4 -2 0 2 4 6 8 10

num

ber

of confs

Q (field theor. cooling tol1)

Field theoretic definition with cooling

5 steps with tol. 5% 10 steps with tol. 1%

Cooling vs. HYP smearing

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 16 / 35

-4

-2

0

2

4

0 20 40 60 80 100 120 140

Q (

field

theor.

coolin

g)

number of cooling iterations

cooling conf 1870cooling conf 1930cooling conf 2090

cooling conf 2240cooling conf 2530cooling conf 2870

-4

-2

0

2

4

6

0 10 20 30 40 50

Q (

field

theor.

HY

Pn)

number of smearing iterations

HYP conf 1600HYP conf 2000

HYP conf 2400HYP conf 2800

HYP conf 3200

Field theoretic definition

cooling HYP smearing

Pairwise comparisons

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 17 / 35

-8

-6

-4

-2

0

2

4

6

8

10

12

2000 2500 3000 3500 4000

Q (

ind

ex n

on

Sm

ea

r s=

0.4

, in

de

x H

YP

1 s

=0

)

conf number

index nonSmear s=0.4

index HYP1 s=0

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

-8 -6 -4 -2 0 2 4 6 8 10 12

Q (

ind

ex H

YP

1 s

=0

)

Q (index nonSmear s=0.4)

correlation=0.950385 (0.007323)index nonSmear s=0.4 + index HYP1 s=0

-8

-6

-4

-2

0

2

4

6

8

10

12

2000 2500 3000 3500 4000

Q (

ind

ex n

on

Sm

ea

r s=

0.4

, sp

ec.

pro

j. n

on

Sm

ea

r)

conf number

index nonSmear s=0.4

spec. proj. nonSmear

-8

-6

-4

-2

0

2

4

6

8

10

12

-8 -6 -4 -2 0 2 4 6 8 10 12

Q (

sp

ec.

pro

j. n

on

Sm

ea

r)

Q (index nonSmear s=0.4)

correlation=0.502528 (0.045219)index nonSmear s=0.4 + spec. proj. nonSmear

0.950(7)

index

HY

P1

index nonSmear

0.503(45)

spec

.pro

j.non

Sm

ear

index nonSmear

Pairwise comparisons

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 18 / 35

-8

-6

-4

-2

0

2

4

6

8

10

12

2000 2500 3000 3500 4000

Q (

ind

ex n

on

Sm

ea

r s=

0.4

, fe

rmio

nic

)

conf number

index nonSmear s=0.4

fermionic

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

-10 -8 -6 -4 -2 0 2 4 6 8 10 12

Q (

ferm

ion

ic)

Q (index nonSmear s=0.4)

correlation=0.472707 (0.045937) index nonSmear s=0.4 + fermionic

-8

-6

-4

-2

0

2

4

6

8

10

12

2000 2500 3000 3500 4000

Q (

ind

ex n

on

Sm

ea

r s=

0.4

, G

F f

low

tim

e t

0)

conf number

index nonSmear s=0.4

GF flow time t0

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

-10 -8 -6 -4 -2 0 2 4 6 8 10 12

Q (

GF

flo

w t

ime

t0

)

Q (index nonSmear s=0.4)

correlation=0.863050 (0.016853) index nonSmear s=0.4 + GF flow time t0

0.473(46)

ferm

ionic

(dis

c.)

index nonSmear

0.863(17)

GF

flow

tim

et 0

index nonSmear

Pairwise comparisons

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 19 / 35

-8

-6

-4

-2

0

2

4

6

8

10

12

2000 2500 3000 3500 4000

Q (

ind

ex n

on

Sm

ea

r s=

0.4

, fie

ld t

he

or.

no

nS

me

ar)

conf number

index nonSmear s=0.4

field theor. nonSmear

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

-10 -8 -6 -4 -2 0 2 4 6 8 10 12

Q (

fie

ld t

he

or.

no

nS

me

ar)

Q (index nonSmear s=0.4)

correlation=0.183216 (0.059880)index nonSmear s=0.4 + field theor. nonSmear

-8

-6

-4

-2

0

2

4

6

8

10

12

1500 2000 2500 3000 3500 4000

Q (

ind

ex H

YP

1 s

=0

, fie

ld t

he

or.

HY

P1

0)

conf number

index HYP1 s=0

field theor. HYP10

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

-10 -8 -6 -4 -2 0 2 4 6 8 10 12

Q (

fie

ld t

he

or.

HY

P1

0)

Q (index HYP1 s=0)

correlation=0.924477 (0.007851) index HYP1 s=0 + field theor. HYP10

0.183(60)

fiel

dth

eor.

non

Sm

ear

index nonSmear

0.924(8)

fiel

dth

eor.

HY

P10

index HYP1

Pairwise comparisons

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 20 / 35

-8

-6

-4

-2

0

2

4

6

8

10

12

1500 2000 2500 3000 3500 4000

Q (

ind

ex H

YP

1 s

=0

, fie

ld t

he

or.

co

olin

g t

ol5

)

conf number

index HYP1 s=0

field theor. cooling tol5

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

-10 -8 -6 -4 -2 0 2 4 6 8 10 12

Q (

fie

ld t

he

or.

co

olin

g t

ol5

)

Q (index HYP1 s=0)

correlation=0.837581 (0.017606)index HYP1 s=0 + field theor. cooling tol5

-3

-2

-1

0

1

2

3

4

5

1500 2000 2500 3000 3500 4000

Q (

sp

ec.

pro

j. n

on

Sm

ea

r, f

erm

ion

ic)

conf number

spec. proj. nonSmear

fermionic

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

Q (

ferm

ion

ic)

Q (spec. proj. nonSmear)

correlation=0.846143 (0.016841) spec. proj. nonSmear + fermionic

0.838(18)

fiel

dth

eor.

cool

.to

l5

index HYP1

0.846(17)

ferm

ionic

(dis

c.)

spec. proj. nonSmear

Pairwise comparisons

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 21 / 35

-8

-6

-4

-2

0

2

4

6

8

10

500 1000 1500 2000 2500 3000 3500 4000

Q (

GF

flo

w t

ime

t0

, G

F f

low

tim

e 3

t0)

conf number

GF flow time t0

GF flow time 3t0

-8

-6

-4

-2

0

2

4

6

8

10

-8 -6 -4 -2 0 2 4 6 8

Q (

GF

flo

w t

ime

3t0

)

Q (GF flow time t0)

correlation=0.961467 (0.004643) GF flow time t0 + GF flow time 3t0

-8

-6

-4

-2

0

2

4

6

8

10

2000 2500 3000 3500 4000

Q (

GF

flo

w t

ime

t0

, S

F H

YP

5 s

=0

.0)

conf number

GF flow time t0

SF HYP5 s=0.0

-8

-6

-4

-2

0

2

4

6

8

10

-8 -6 -4 -2 0 2 4 6 8

Q (

SF

HY

P5

s=

0.0

)

Q (GF flow time t0)

correlation=0.922854 (0.008417) GF flow time t0 + SF HYP5 s=0.0

0.961(5)

GF

flow

tim

e3t

0

GF flow time t0

0.923(8)

SF

HY

P5s

=0

GF flow time t0

Pairwise comparisons

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 22 / 35

-8

-6

-4

-2

0

2

4

6

8

2000 2500 3000 3500 4000

Q (

GF

flo

w t

ime

t0

, fie

ld t

he

or.

HY

P1

0)

conf number

GF flow time t0

field theor. HYP10

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

Q (

fie

ld t

he

or.

HY

P1

0)

Q (GF flow time t0)

correlation=0.970876 (0.003396) GF flow time t0 + field theor. HYP10

-8

-6

-4

-2

0

2

4

6

8

10

2000 2500 3000 3500 4000

Q (

GF

flo

w t

ime

3t0

, fie

ld t

he

or.

co

olin

g t

ol1

)

conf number

GF flow time 3t0

field theor. cooling tol1

-8

-6

-4

-2

0

2

4

6

8

10

-8 -6 -4 -2 0 2 4 6 8

Q (

fie

ld t

he

or.

co

olin

g t

ol1

)

Q (GF flow time 3t0)

correlation=0.982992 (0.003395)GF flow time 3t0 + field theor. cooling tol1

0.971(3)

fiel

dth

eor.

HY

P10

GF flow time t0

0.983(3)

fiel

dth

eor.

cool

.to

l1

GF flow time 3t0

Pairwise comparisons

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 23 / 35

-8

-6

-4

-2

0

2

4

6

8

10

1500 2000 2500 3000 3500 4000

Q (

fie

ld t

he

or.

HY

P1

0,

fie

ld t

he

or.

HY

P4

0)

conf number

field theor. HYP10

field theor. HYP40

-8

-6

-4

-2

0

2

4

6

8

10

-8 -6 -4 -2 0 2 4 6 8 10

Q (

fie

ld t

he

or.

HY

P4

0)

Q (field theor. HYP10)

correlation=0.986610 (0.001899) field theor. HYP10 + field theor. HYP40

-8

-6

-4

-2

0

2

4

6

8

10

1500 2000 2500 3000 3500 4000

Q (

fie

ld t

he

or.

HY

P4

0,

fie

ld t

he

or.

co

olin

g t

ol5

)

conf number

field theor. HYP40

field theor. cooling tol5

-8

-6

-4

-2

0

2

4

6

8

10

-8 -6 -4 -2 0 2 4 6 8 10

Q (

fie

ld t

he

or.

co

olin

g t

ol5

)

Q (field theor. HYP40)

correlation=0.940834 (0.007505)field theor. HYP40 + field theor. cooling tol5

0.987(2)

fiel

dth

eor.

HY

P40

field theor. HYP10

0.941(8)

fiel

dth

eor.

cool

.to

l5

field theor. HYP40

Pairwise comparisons

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 24 / 35

-8

-6

-4

-2

0

2

4

6

8

10

1500 2000 2500 3000 3500 4000

Q (

fie

ld t

he

or.

co

olin

g t

ol5

, fie

ld t

he

or.

co

olin

g t

ol1

)

conf number

field theor. cooling tol5

field theor. cooling tol1

-8

-6

-4

-2

0

2

4

6

8

10

-8 -6 -4 -2 0 2 4 6 8 10

Q (

fie

ld t

he

or.

co

olin

g t

ol1

)

Q (field theor. cooling tol5)

correlation=0.939104 (0.008230)field theor. cooling tol5 + field theor. cooling tol1

-10

-5

0

5

10

15

1500 2000 2500 3000 3500 4000

Q (

SF

HY

P1

s=

0.0

, S

F H

YP

1 s

=0

.75

)

conf number

SF HYP1 s=0.0

SF HYP1 s=0.75

-10

-5

0

5

10

15

-10 -8 -6 -4 -2 0 2 4 6 8 10 12

Q (

SF

HY

P1

s=

0.7

5)

Q (SF HYP1 s=0.0)

correlation=0.907585 (0.009311) SF HYP1 s=0.0 + SF HYP1 s=0.75

0.939(8)

fiel

dth

eor.

cool

.to

l1

field theor. cool. tol5

0.908(9)

SF

HY

P1s

=0.

75

SF HYP1 s = 0

Pairwise comparisons

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 25 / 35

-8

-6

-4

-2

0

2

4

6

8

10

12

1500 2000 2500 3000 3500 4000

Q (

SF

HY

P1

s=

0.0

, S

F H

YP

5 s

=0

.0)

conf number

SF HYP1 s=0.0

SF HYP5 s=0.0

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

-10 -8 -6 -4 -2 0 2 4 6 8 10 12

Q (

SF

HY

P5

s=

0.0

)

Q (SF HYP1 s=0.0)

correlation=0.928337 (0.006411) SF HYP1 s=0.0 + SF HYP5 s=0.0

-10

-8

-6

-4

-2

0

2

4

6

8

10

1500 2000 2500 3000 3500 4000

Q (

SF

HY

P5

s=

0.0

, S

F H

YP

5 s

=0

.5)

conf number

SF HYP5 s=0.0

SF HYP5 s=0.5

-10

-8

-6

-4

-2

0

2

4

6

8

10

-8 -6 -4 -2 0 2 4 6 8 10

Q (

SF

HY

P5

s=

0.5

)

Q (SF HYP5 s=0.0)

correlation=0.994330 (0.000735) SF HYP5 s=0.0 + SF HYP5 s=0.5

0.928(6)

SF

HY

P5s

=0

SF HYP1 s = 0

0.994(1)

SF

HY

P5s

=0.

5

SF HYP5 s = 0

Pairwise comparisons

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 26 / 35

-8

-6

-4

-2

0

2

4

6

8

10

1500 2000 2500 3000 3500 4000

Q (

SF

HY

P5

s=

0.0

, fie

ld t

he

or.

HY

P1

0)

conf number

SF HYP5 s=0.0

field theor. HYP10

-8

-6

-4

-2

0

2

4

6

8

10

-8 -6 -4 -2 0 2 4 6 8 10

Q (

fie

ld t

he

or.

HY

P1

0)

Q (SF HYP5 s=0.0)

correlation=0.972357 (0.003023) SF HYP5 s=0.0 + field theor. HYP10

-4

-3

-2

-1

0

1

2

3

4

5

2000 2500 3000 3500 4000

Q (

sp

ec.

pro

j. n

on

Sm

ea

r, s

pe

c.

pro

j. H

YP

1)

conf number

spec. proj. nonSmear

spec. proj. HYP1

-3

-2

-1

0

1

2

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

Q (

sp

ec.

pro

j. H

YP

1)

Q (spec. proj. nonSmear)

correlation=-0.050581 (0.049186) spec. proj. nonSmear + spec. proj. HYP1

0.972(3)

fiel

dth

eor.

HY

P10

SF HYP5 s = 0

-0.051(49)

spec

.pro

j.H

YP

1

spec. proj. nonSmear

Table of correlations

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 27 / 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 1.000(0) 0.950(7) 0.877(14) 0.951(4) 0.929(9) 0.925(10) 0.506(44) 0.474(49) 0.863(16) 0.827(21) 0.813(20) 0.183(59) 0.907(12) 0.888(14) 0.703(33) 0.640(41) 0.830(21) 0.810(22)2 0.950(7) 1.000(0) 0.897(10) 0.990(2) 0.937(6) 0.933(6) 0.474(46) 0.462(50) 0.863(16) 0.820(21) 0.804(23) 0.167(43) 0.924(8) 0.906(9) 0.667(34) 0.585(36) 0.838(18) 0.808(20)3 0.877(14) 0.897(10) 1.000(0) 0.907(9) 0.854(12) 0.847(12) 0.404(50) 0.394(49) 0.774(26) 0.734(30) 0.718(32) 0.182(40) 0.834(15) 0.813(16) 0.652(32) 0.550(40) 0.752(23) 0.726(26)4 0.951(4) 0.990(2) 0.907(9) 1.000(0) 0.934(6) 0.928(6) 0.474(46) 0.452(52) 0.865(16) 0.821(22) 0.810(22) 0.172(44) 0.919(8) 0.900(10) 0.664(33) 0.580(40) 0.836(16) 0.806(20)5 0.929(9) 0.937(6) 0.854(12) 0.934(6) 1.000(0) 0.994(0) 0.493(41) 0.457(46) 0.915(9) 0.868(15) 0.853(17) 0.151(44) 0.967(3) 0.945(5) 0.647(34) 0.572(36) 0.879(11) 0.835(15)6 0.925(10) 0.933(6) 0.847(12) 0.928(6) 0.994(0) 1.000(0) 0.490(43) 0.455(45) 0.924(8) 0.877(15) 0.859(17) 0.154(46) 0.972(2) 0.951(5) 0.644(35) 0.575(36) 0.889(11) 0.842(15)7 0.506(44) 0.474(46) 0.404(50) 0.474(46) 0.493(41) 0.490(43) 1.000(0) 0.847(16) 0.620(39) 0.639(37) 0.647(38) 0.074(47) 0.556(42) 0.547(41) 0.464(44) 0.562(47) 0.556(41) 0.600(37)8 0.474(49) 0.462(50) 0.394(49) 0.452(52) 0.457(46) 0.455(45) 0.847(16) 1.000(0) 0.545(45) 0.558(42) 0.568(38) 0.051(42) 0.517(49) 0.502(50) 0.452(47) 0.534(41) 0.508(43) 0.553(40)9 0.863(16) 0.863(16) 0.774(26) 0.865(16) 0.915(9) 0.924(8) 0.620(39) 0.545(45) 1.000(0) 0.976(3) 0.962(4) 0.153(58) 0.971(3) 0.967(4) 0.663(39) 0.640(44) 0.974(4) 0.954(7)

10 0.827(21) 0.820(21) 0.734(30) 0.821(22) 0.868(15) 0.877(15) 0.639(37) 0.558(42) 0.976(3) 1.000(0) 0.992(1) 0.132(60) 0.934(9) 0.937(8) 0.632(43) 0.614(44) 0.965(6) 0.978(4)11 0.813(20) 0.804(23) 0.718(32) 0.810(22) 0.853(17) 0.859(17) 0.647(38) 0.568(38) 0.962(4) 0.992(1) 1.000(0) 0.138(57) 0.919(9) 0.921(9) 0.624(43) 0.615(46) 0.952(7) 0.983(3)12 0.183(59) 0.167(43) 0.182(40) 0.172(44) 0.151(44) 0.154(46) 0.074(47) 0.051(42) 0.153(58) 0.132(60) 0.138(57) 1.000(0) 0.170(44) 0.166(46) 0.256(37) 0.171(39) 0.178(45) 0.124(45)13 0.907(12) 0.924(8) 0.834(15) 0.919(8) 0.967(3) 0.972(2) 0.556(42) 0.517(49) 0.971(3) 0.934(9) 0.919(9) 0.170(44) 1.000(0) 0.987(1) 0.653(36) 0.611(33) 0.938(8) 0.904(11)14 0.888(14) 0.906(9) 0.813(16) 0.900(10) 0.945(5) 0.951(5) 0.547(41) 0.502(50) 0.967(4) 0.937(8) 0.921(9) 0.166(46) 0.987(1) 1.000(0) 0.629(35) 0.589(35) 0.941(7) 0.908(11)15 0.703(33) 0.667(34) 0.652(32) 0.664(33) 0.647(34) 0.644(35) 0.464(44) 0.452(47) 0.663(39) 0.632(43) 0.624(43) 0.256(37) 0.653(36) 0.629(35) 1.000(0) 0.776(22) 0.587(39) 0.580(40)16 0.640(41) 0.585(36) 0.550(40) 0.580(40) 0.572(36) 0.575(36) 0.562(47) 0.534(41) 0.640(44) 0.614(44) 0.615(46) 0.171(39) 0.611(33) 0.589(35) 0.776(22) 1.000(0) 0.573(35) 0.581(37)17 0.830(21) 0.838(18) 0.752(23) 0.836(16) 0.879(11) 0.889(11) 0.556(41) 0.508(43) 0.974(4) 0.965(6) 0.952(7) 0.178(45) 0.938(8) 0.941(7) 0.587(39) 0.573(35) 1.000(0) 0.939(8)18 0.810(22) 0.808(20) 0.726(26) 0.806(20) 0.835(15) 0.842(15) 0.600(37) 0.553(40) 0.954(7) 0.978(4) 0.983(3) 0.124(45) 0.904(11) 0.908(11) 0.580(40) 0.581(37) 0.939(8) 1.000(0)

1 = index nonSmear s = 0.4, 2 = index HYP1 s = 0, 3 = SF HYP1 s = 0.75,4 = SF HYP1 s = 0, 5 = SF HYP5 s = 0.5, 6 = SF HYP5 s = 0,7 = spectral projectors nonSmear, 8 = fermionic (from disconnected loops),9 = GF flow time t0, 10 = GF flow time 2t0, 11 = GF flow time 3t0,12 = field theor. nonSmear, 13 = field. theor. HYP10, 14 = field. theor. HYP40,15 = field. theor. APE10, 16 = field. theor. APE40, 17 = field theor. cooling tol. 5%,

18 = field theor. cooling tol. 1%

Plot of correlations

Presentationoutline

Introduction

Results

MC histories

Histograms

Cool. vs. HYP

Pair comparisons

Correlations

Topo. susc.

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 28 / 35

index nonSmear s=0.4

index nonSmear s=0

index HYP1 s=0

SF HYP1 s=0.75

SF HYP1 s=0.0

SF HYP5 s=0.5

SF HYP5 s=0.0

spec. proj. nonSmear

spec. proj. HYP1

fermionic

GF flow time t0

GF flow time 2t0

GF flow time 3t0

field theor. nonSmear

field theor. HYP10

field theor. HYP40

field theor. APE10

field theor. APE40

field theor. cooling tol5

field theor. cooling tol1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

me

tho

d 2

method 1

-0.2

0

0.2

0.4

0.6

0.8

1

Correlation towards the continuum limit

Presentationoutline

Introduction

Results

MC histories

Histograms

Cool. vs. HYP

Pair comparisons

Correlations

Topo. susc.

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 29 / 35

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

corr

ela

tion

a/r0

index HYP1 s=0 vs. spec. proj. nonSmearindex HYP1 s=0 vs. GF flow time t0

index HYP1 s=0 vs. field theor. cooling tol1index HYP1 s=0 vs. field theor. cooling tol5

spec. proj. nonSmear vs. GF flow time t0spec. proj. nonSmear vs. field theor. cooling tol1spec. proj. nonSmear vs. field theor. cooling tol5

GF flow time t0 vs. field theor. cooling tol1GF flow time t0 vs. field theor. cooling tol5

field theor. cooling tol1 vs. field theor. cooling tol5

Topological susceptibility – b40.16

Presentationoutline

Introduction

Results

MC histories

Histograms

Cool. vs. HYP

Pair comparisons

Correlations

Topo. susc.

Conclusions

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 30 / 35

0

1e-05

2e-05

3e-05

4e-05

5e-05

6e-05

7e-05

8e-05

9e-05

0.0001

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

a4 χ

ren

definition ID

noSmears=0.4

noSmears=0

HYP1s=0

HYP1s=0

HYP1s=0.75

HYP5s=0

HYP5s=0.5

noSmear

HYP1

noSmear

flow time: t0 2t0 3t0

HYP1

HYP2

HYP5

HYP10

HYP20

HYP30

HYP40

HYP50

tol.5

tol.1

indexspectral flow

spec. projectorsfermionic (disc.)

field theor. noSmearfield theor. GF

field theor. HYPfield theor. cool.

Topological susceptibility

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 31 / 35

used only pion masses mπ ≤ 400 MeV

spectral projectors: r0Σ1/3 = 0.646(59) fermionic: r0Σ1/3 = 0.644(20)

Tree-level χPT fit: r40χ =r3

0Σ·r0µR

2

compare to direct determination[K.C., E. García Ramos, K. Jansen, JHEP 10(2013)175]

r0Σ1/3cont,Nf=2+1+1 = 0.680(29)

Topological susceptibility

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 32 / 35

used all pion masses (mπ = [260, 500] MeV)

spectral projectors: r0Σ1/3 = 0.612(57) fermionic: r0Σ1/3 = 0.622(19)

Tree-level χPT fit: r40χ =r3

0Σ·r0µR

2

compare to direct determination[K.C., E. García Ramos, K. Jansen, JHEP 10(2013)175]

r0Σ1/3cont,Nf=2+1+1 = 0.680(29)

My subjective ranking of costs

Presentationoutline

Introduction

Results

Conclusions

Costs

Cleanness

Which definition?

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 33 / 35

From the most expensive to the cheapest:

1. index of the overlap Dirac operator

2. spectral flow

3. spectral projectors

4. fermionic from disconnected loops (but...)

5. field theoretic (with smearing or cooling)

6. field theoretic with gradient flow

My subjective ranking

of theoretical cleanness

Presentationoutline

Introduction

Results

Conclusions

Costs

Cleanness

Which definition?

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 34 / 35

From the cleanest to the problematic ones:

1. index of the overlap Dirac operatorspectral flowfield theoretic with gradient flow

2. spectral projectors (fully clean for χ)

3. field theoretic with smearing

4. field theoretic with cooling

5. fermionic from disconnected loops

Which definition to use?

Presentationoutline

Introduction

Results

Conclusions

Costs

Cleanness

Which definition?

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 35 / 35

The answer to this question is not unique:

• for the computation of topological susceptibility:spectral projectors, field theoretic with gradient flow

• for sorting configurations:field theoretic

• for weighting results with the topological charge:field theoretic

And the winner is: ,

field theoretic with gradient flow

Thank you for your attention!

Presentationoutline

Introduction

Results

Conclusions

Thank you for yourattention!

Krzysztof Cichy ETMC Meeting Grenoble – 26-28 May 2014 – 36 / 35