Intro Mass terms Index Spectrum Meson Overlap

Adams and Hoelbling fermions:numerical properties
Philippe de Forcrand
ETH Zürich & CERN & YITP
with Aleksi Kurkela (McGill) & Marco Panero (Helsinki)
arXiv:1202.1867, Feb. 9, 2012

Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson Overlap

Motivation• Light u , d quarks needed to simulate correct physics → expensive• Cost-saving: staggered fermions(1/4 d.o.f.)

SF = ∑x χ̄(x ) ∑µ ηµ (x )(Uµ (x )χ(x + µ̂) − Uµ† (x − µ̂)χ(x − µ̂)) + mq ∑x χ̄(x )χ(x )ηµ = ±1; {γµ , γν } = 2δµν → ∏4 η = −1 around any plaquettep• Drawback: Nf = 4 (degenerate when a = 0) “tastes” → det(Dst )“rooting is evil”

Mike Creutz- non-locality?
- ’t Hooft vertex, U (1)A breaking?
- staggered fermions “don’t feel the topology”• No quartet of low-lying eigenvalues↔

no index theorem• Adams to the rescue: Nf = 2 staggered overlap fermions0912.2850, 1008.28331009.5362• Hoelbling: Nf = 1
Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson Overlap

Construction

Two ingredients:
1: taste-dependent mass term (cf. Wilson term)=⇒ split tastes into several branches – fine-tune for massless limit
2: plug into overlap (cf. Neuberger)=⇒ eliminate fine-tuning, etc.. – large computational overhead

Look at 1 then 2

Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson Overlap

Possible mass terms

Golterman & Smit (1984)
10-linkγµ 1-linkγµ γν 2-linkγµ γν γρ 3-link
γ1 γ2 γ3 γ4 4-link• 1-link and 3-link do not satisfy γ5 -hermiticity → determinant complex- can be fixed with 2 conjugate copies, ie. doubling Nf- 1-link: cancellations between kinetic & mass terms
free spectrum on a circle!
0-link = KS; 2-link = Hoelbling; 4-link = Adams
Notation: 0-link → m, 2-link → MHoelbling , 4-link → MAdams ∝ Γ55 Γ5• NB. Also can multiply above mass terms by Γ55 ≡ (−)x ?
Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermionsFree spectra and taste reduction ID/ stD/ st + MAdams

2D/ st + MHoelbling

22

1

11.5
10

Im[λ]

0.5
Im[λ]

Im[lambda]

Intro Mass terms Index Spectrum Meson Overlap

00-0.5-1-1-1-1.5-2-2-2-1-0.5
00.5
Re[lambda]

Nf = 4Ph. de Forcrand

1-1-0.50Re[λ]/ρ

0.5

Nf = 2 + 2

YITP, Feb. 2012

1-2-1.5-1-0.50Re[λ]/ρ

0.5

Nf = 1 + 2 + 1Adams & Hoelbling fermions

11.5

2

Intro Mass terms Index Spectrum Meson Overlap

Free spectra and taste reduction II• Select one branch by x-shift, ie. additive mass renormalization (cf. Wilson)
Other branch(es) have masses O (1/a), ie. doublers
22

11Heavy modes
10Im[λ]ψ Γ5Γ55ψ

Im[λ]

0.5

00-0.5Light modes-1-1-1-π/2-10max(p)1π/2-2-2-1.5-1-0.50Re[λ]/ρ

0.5

11.5

2-2-1-0.50Re[λ]/ρ

0.51• Real (low-momentum) modes are taste-chiral in each branch → Γ55 Γ5 ≈ ±1
⇒ can trade Γ5 = γ5 ⊗ 1 + O (a) for Γ55 = γ5 ⊗ γ5 exactly → index theorem
Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson Overlap

Index from eigenvalue flow
• Index from flow of eigenvalues λ(m) of H (m) = γ5 (D/ + m)
• (D/ + m0 ) has zero-mode |Ψ0 i → eigenvalue λ(m0 ) = 0 for H (m0 )• Perturb m away from m0 → eigenvalue displaced by hΨ0 |γ5 (m − m0 )|Ψ0 i
λ(m) = ±(m − m0 ) (ie. crossing) IF hΨ0 |γ5 |Ψ0 i = ±1, ie. |Ψ0 i chiral• Works with D/ =D/ Wilson , or D
/ st + MAdams , D/ st + MHoelbling with γ5 → Γ55• Alternative: can also consider flow for Ĥ (ρ) = Γ55 (D/ st + ρMAdams )Adams’ original proposal•D/ st has near-chiral, near-zero modes• MAdams = Γ55 Γ5 → perturbation ρΓ5 ≈ ργ5 ⊗ 1“Classic” versus “original” ?
Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson Overlap

Comparing two eigenvalue flows• Adams’ original
22

1.5

1.5

1

1

0.4

0.2
0.5

0.5
Adams 163x32 cold

0Adams 84 Q=10-0.5

Adams 124 b=6.0

0-0.5
-0.2

-1-1-1.5-1.5-2-0.4-2-4-3-2

-1

0

12

34-4-3-2-1

0

12

34-4-3-2-10m

123-4-3-2-10

12

34• Adams’ classic
2221.5

1.5

1.5

11

10.5

0.5

0.5

000-0.5

-0.5

-0.5

-1-1-1-1.5

-1.5

-1.5

-2-2-4-3-2-1

0

123

3

Cold config. 16 x32
Ph. de Forcrand

4-2
-4

-3-2

-1

0

1

2

Cooled Q = 1, 8YITP, Feb. 201234

4

Q = −1, β = 6.0, 12Adams & Hoelbling fermions44

Intro Mass terms Index Spectrum Meson Overlap

Index theorem• Cold configuration: agreement with analytic result
222

1.5

1.5

1.5

11

1

0.5

0.5

0.5000-0.5

-0.5

-0.5

-1-1-1-1.5

-1.5

-1.5

-2-2-4-3-2-101Adams

234Neuberger 163x32 cold-2-4-3-2-10

1Hoelbling234-4-3-2-1

0

Neuberger

12• Cooled Q = 1 instanton: Nf × Q crossings
22

1.5

1.5

24

Hoelbling 8 Q=1
1.5

1

1

1

0.5

0.5

0.5

0

00-0.5

-0.5

-0.5

-1

-1-1

-1.5

-1.5

-1.5

-2-2-4-3-2-10

1Adams

Ph. de Forcrand234

Neuberger 84 Q=1-2-4-3-2-1

01

Hoelbling

YITP, Feb. 2012

234-4

Adams & Hoelbling fermions-3-2-10Neuberger

12Intro Mass terms Index Spectrum Meson Overlap

Effects of gauge field fluctuations• β = 6.0, Q = 1: eigenvalue gap closes, esp. Adams
24

Hoelbling 12 b=6.0
1.5

0.4

0.4

0.2

0.2

00

-0.2

-0.2

1
0.5
0

Neuberger 124 b=6.0

-0.5
-1
-1.5

-0.4

-0.4

-2-4-3-2-1

01Adams

Ph. de Forcrand234-4-3-2-101234-4

Hoelbling

YITP, Feb. 2012

Adams & Hoelbling fermions-3-2m-1

Neuberger01

Intro Mass terms Index Spectrum Meson Overlap

Effects of gauge field fluctuations• The width of the spectrum fluctuates (shrinks)→ fine-tuning for massless quarks• The gap in the spectrum fills up→ distinction between light modes and doublers blurred

Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson Overlap

Eigenvalue spectra: Adams vs Hoelbling, 84

10-1-2-0.50Re[λ]/ρ

4

1.5
110.5

0.5

0.5

000-0.5-0.5-1-1-1-1.5-1.5-1.5

free 32-2-0.502.50-10.5

β = 6.0

1-2-1-0.5

00.5

1-1

β = 6.0, vary ρ
2.52-link, beta=6.0

2

1

1.5-0.5-184, beta=5.8
rho=1

2

1

10.584, beta=6.0
rho=1
rho=2
rho=3

2

1.5-2-1

Im[λ]

Im[λ]

84, beta=6.0
rho=12222

1.5

1.5

11

10.5

0.5

0.5

000-0.5-0.5-0.5-1-1-1-1.5-1.5-1.5-2-2-2.5

00.5

β = 5.8

2.52-link, beta=6.02
1.5-2.5-0.5

12-link, beta=5.8-2-2-2-1.5-1-0.50Re[λ]/ρ

0.5

Ph. de Forcrand

1

1.5

2-2-1

0

1YITP, Feb. 2012

2-2.5-2-1012Adams & Hoelbling fermions

-2-1

012

Intro Mass terms Index Spectrum Meson Overlap

Usefulness
Superior robustness of Hoelbling fermions:
2-link vs 4-link transporters• Can always smear links to suppress fluctuations: Thanks Stephan Dürr
4D: N_c=3, L=4, β=5.6, |q|=1

4D: N_c=3, L=4, β=5.6, |q|=1

4D: N_c=3, L=4, β=5.6, |q|=1

22

2

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0

0

0

−0.5

−0.5

−0.5

−1−1adams, n_smear=0adams, n_smear=1adams, n_smear=3adams, n_smear=7stagger, n_smear=7−1.5

−2

00.5

1

1.5

2

2.5

3Adams

3.5

4

4.5

−1hoelb3, n_smear=0hoelb3, n_smear=1hoelb3, n_smear=3hoelb3, n_smear=7stagger, n_smear=7−1.5

5−200.5

11.5

22.5

33.5

44.5wils, n_smear=0wils, n_smear=1wils, n_smear=3wils, n_smear=7−1.5

5−20

Hoelbling
44 , β = 5.6
Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermions

0.5

1

1.5

2

2.5

3

3.5

Wilson44.5

5

Intro Mass terms Index Spectrum Meson Overlap

Pion mass• Compute quark propagator G(x , y , z , t ) from point source:- 163 × 32 lattices, β = 6.0 quenched- Form ~p = ~0 meson correlator (connected diagram only)
C (t ) = ∑xyz G(x , y , z , t )Γ55 G(x , y , z , t )† Γ55 = ∑xyz |G(x , y , z , t )|2- Look for effective mass plateau → (lightest) pion mass (amπ )• Monitor (amπ )2 vs (amq ): PCAC + additive mass renormalization

Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson Overlap

Pion correlator• Free field:
standard KS, free

Adams kernel, free100000

2-link mass, free100

110.0110.01

0.00011e-05
0.0001
1e-06
1e-10

1e-06
1e-08
1e-08

1e-15

1e-20

1e-25

m=0.125m=0.25
m=0.5
m=1.0
m=1.5
m=1.75m=1.875
02

1e-10
1e-12
1e-14
4

6810

1214KSmπ = 0 for m0 = 0Ph. de Forcrand16

m=0.0m=0.125
m=0.25m=0.375
m=0.5
m=0.625m=0.75m=0.875m=1.00

1e-10
1e-12
1e-14
24

6810Adams
m0 = 1YITP, Feb. 2012

121416m=0.125m=0.25m=0.5m=0.75m=1.0
m=1.25m=1.5
m=1.75m=2.00Adams & Hoelbling fermions

2468

1012Hoelbling
m0 = 2 and 01416

Intro Mass terms Index Spectrum Meson Overlap

Pion mass vs mq
3• β = 6.0, 16× 32 quenched:
4link pion mass vs m

2link pion mass vs mq

q

2

2
b=6.0

b=6.0

1.5

1.5

1

1

0.5

0.5

0

0
0

0.2

0.4

0.6

0.8

1Adamsm0 ∼ 0.25 instead of 1

00.5

14 links vs 2 links: 0.25/1 ∼ (1.15/2)2
Use Hoelbling fermions for Nf = 2: no need to tune m0 !
Ph. de Forcrand

YITP, Feb. 2012

1.52Hoelbling
sqrt behaviour for Nf = 2 and Nf = 1 ??
m0 ∼ 1.15 instead of 2 or 0Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson Overlap

Pion correlator Hoelbling• β = 6.0, 163 × 32 quenched:

2-link mass, beta=6.0

100
1
0.01
0.0001
1e-06
1e-08
1e-10
1e-12
1e-14

m=0.125m=0.25m=0.5
m=0.75m=1.0
m=1.25m=1.5
m=1.75m=2.00Ph. de Forcrand246YITP, Feb. 2012

8101214

Adams & Hoelbling fermions16

Intro Mass terms Index Spectrum Meson Overlap

Construction Free field Locality Cost Inversion

Overlap construction• Just like Neuberger: Dsov = 1 + “γ005 sign(H (−m0 ))with “γ005 = Γ55 = (−)x +y +z +t(need “γ005 2 = 1)

• Potential advantages:- cheaper (4 times fewer d.o.f. per site)- more robust ?

And reduces Nf = 4 to Nf = 2 tastes without fine-tuning

N.B. m0 is really ρ in Adams’ proposal [no mass shift]

Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson OverlapConstruction Free field Locality Cost Inversion

Free field: Uµ (x ) = 1 ∀x, µSpectrum of kernel: γ5 HW (m0 = −1) and γ5 HAdams (m0 = −1)
2

Im[λ]

10-1-20

246Re[λ]/r

γ5 sign(H ) = √ D† projects eigenvalues of D = γ5 H on unit circle
D D

Adams: two low-p eigenmodes projected to -1, two projected to +1 ⇒ Nf = 2
Ph. de Forcrand

YITP, Feb. 2012Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson Overlap

Construction Free field Locality Cost Inversion

Locality of operator?
Maxy |Mxy | versus |x − y | (Manhattan distance)
Adams
1Neuberger1

beta=6.0
beta=6.2
beta=6.4

0.1

cf. hep-lat/9808010

Neuberger beta=6.0
beta=6.2
beta=6.4

0.1

0.01

0.01

0.001

0.001

0.00010.0001

1e-05

1e-05

1e-06

1e-06

1e-07

1e-07
0

51015202505101520

Adams comparable to Neuberger although kernel less local (4-link)

Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermions

25Intro Mass terms Index Spectrum Meson Overlap

Construction Free field Locality Cost Inversion

Locality of operator?
Maxy |Mxy | versus |x − y | (Manhattan distance)

Both, β = 6.0

Both, cold
1cf. hep-lat/9808010

1Adams 124 free
Neuberger 124 free

0.1

0.1

0.01

0.01

0.001

0.001

0.00010.0001

1e-05

1e-05

1e-06

1e-06

1e-07

Adams beta=6.0
Neuberger beta=6.0

1e-07
0510152025

05101520

Adams comparable to Neuberger although kernel less local (4-link)

Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermions

25Intro Mass terms Index Spectrum Meson Overlap

Construction Free field Locality Cost Inversion

Cost of applying operator• Multiplication by D: about 2 times faster for Adams (no Dirac indices)• Sign(H) [using CG, no deflation]:- about 8 times faster for Adams on easy cases- about 2-3 times faster on hard cases
Can optimize m0 and ρ in Adams’ operator: not exploited yetAlso:
improved kinetic operator, link smearing (kinetic and/or mass),
deflation, preconditioning, ...

Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson Overlap

Construction Free field Locality Cost Inversion

Cost of inversion: compare with Neuberger
Apples with apples:- same gauge field (124 , β = 6.0)
- same basic algorithm (CG inner, CG outer)
Adams versus Neuberger• β = 6.011

44

Adams CG 12 b=6.0 m=0.1
4Neuberger CG 12 b=6.0 m=0.1

10.1

Adams CG 12 b=6.0 m=0.1
4Neuberger CG 12 b=6.0 m=0.1
0.1

0.01

0.01

0.01

0.001

0.001

0.001

0.00010.00010.0001

1e-05

1e-05

1e-05

1e-06

1e-06

1e-06

1e-07

1e-07

1e-08

1e-07

1e-08
01020304050604Adams CG 12 b=6.0 m=0.1
4Neuberger CG 12 b=6.0 m=0.1

0.1

7080

1e-08
020000

40000

60000

Nb. CG iterations

80000

100000 120000 140000 160000 180000 200000

0Nb. MxVOuter CG iter.

MxV

Net CPU gain: factor 2-3 over Neuberger...
Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermions200040006000
CPU time

CPU80001000012000

Intro Mass terms Index Spectrum Meson Overlap

Construction Free field Locality Cost Inversion

Cost of inversion: compare with Neuberger
Adams versus Neuberger• β = 6.0
114

14Adams CG 12 b=6.0 m=0.1
4Neuberger CG 12 b=6.0 m=0.1

Adams CG 12 b=6.0 m=0.1
4Neuberger CG 12 b=6.0 m=0.1

0.1

0.1

0.01

0.01

0.01

0.001

0.001

0.001

0.00010.00010.0001

1e-05

1e-05

1e-05

1e-06

1e-06

1e-06

1e-07

1e-07

1e-08

1e-07

1e-08
010203040

50

6070801e-08
020000

40000

60000

Nb. CG iterations

80000

100000 120000 140000 160000 180000 200000

01

Adams CG 1244 free m=0.1
Neuberger CG 12 free m=0.1

1

Adams CG 1244 free m=0.1
Neuberger CG 12 free m=0.1

0.01

0.01

0.01

0.001

0.001

0.00010.00010.0001

1e-05

1e-05

1e-05

1e-06

1e-06

1e-06

1e-07

1e-07

202512000

Adams CG 1244 free m=0.1
Neuberger CG 12 free m=0.1

1e-08
02000

4000

6000
8000
Nb. MxV10000

12000

14000

0100

• Free field: now factor 8+ → try to keep the free spectrum
Ph. de Forcrand

10000

1e-07

1e-08
1015Nb. CG iterations

8000

0.10.001

1e-08

6000
CPU time

CPU0.1

54000

MxV0.102000

Nb. MxVOuter CG iter.
14Adams CG 12 b=6.0 m=0.1
4Neuberger CG 12 b=6.0 m=0.1

0.1

YITP, Feb. 2012

Adams & Hoelbling fermions

200300400
CPU time

500

600

700

Intro Mass terms Index Spectrum Meson Overlap

Construction Free field Locality Cost Inversion

Conclusions• Index theorem, overlap etc.. work as advertised• Compare taste-dependent mass terms exhaustively first – then tryoverlap• Here: 2-link (Hoelbling) more robust than 4-link (Adams)• 2-link → Nf = 2 without fine-tuning ?
• Other possibilities not yet considered?
eg. start with 8 tastes, split 1-3-3-1

Ph. de Forcrand

YITP, Feb. 2012

Adams & Hoelbling fermionsIntro Mass terms Index Spectrum Meson Overlap

Construction Free field Locality Cost Inversion

Sign problem?

Thanks Mike, Tatsu, TaroTake Q = 1 configuration → 1+2+1 real eigenvalues.
21110••••-10••••N-1-2-1.5-1-0.50Re[λ]/ρ

0.5

11.52••••

N-2-2-1.5-1det > 0Ph. de Forcrand

0N-1-2-2Im[λ]2Im[λ]Im[λ]N

2Vary m0 :-0.50Re[λ]/ρ

0.5

11.5

2-2

det < 0YITP, Feb. 2012

Adams & Hoelbling fermions-1.5-1-0.50Re[λ]/ρ

0.5

det < 01

1.52