Correlation fun tions

x

(S(~x, t;~0, 0) S ^† (~x, t;~0, 0)).

^(B.40)

Weintrodu e the Fourier transformof the positionspa e propagator:

S(~x, t;~0, 0) = 1

where

N

^and

N 4

^are ^the ^number ^of ^latti
e ^sites ⁱⁿ ^the ^spatial ^and ^temporal

dire tions, respe tively. This yields:

C P P (t) = 1

Using the following expression forthe Dira -deltafun tion:

δ(~p − ~p ^′ ) = 1

S(p) =

4

^or

5 X

ξ=0

S _ξ (p)γ _ξ ,

^(B.45)

where

γ 0 ≡

¹ ^and ^the ^index

ξ

^runs ^from ⁰ ^to ⁴ ⁱⁿ ^the ^ase ^of ^overlap ^and

Creutz fermions orfrom 0to 5in the ase of Wilson twistedmass fermions.

Hen e, we obtain:

C P P (t) = N c N d

N ³ N ₄ ² X

~ p

X

p 4 ,p ^′ ₄ 4

^or

5 X

ξ=0

S ξ (~p, p 4 ) S _ξ ^∗ (~p, p ^′ ₄ ))e ^i(p ⁴ ^−p ^′ ⁴ ^)t ,

^(B.46)

where

N d =

^T^r

(

)

^is^the ^number^of ^Dira^omponents ^(i.e. ^the ^dimension ^of

spa e-time) and

N c =

^Tr

(

c )

^is ^the ^number ^of ^olours ⁽ⁱⁿ ^the ^free ^ase ^the

stru ture in olour spa e istrivial).

Improvements of the HMC

algorithm

The gaugeeld ongurations that wehave usedforthis proje t were

gener-atedwiththetwistedmassLatti eQCDprogramsuite(tmLQCD)ofJansen

and Urba h [83℄. A detailed des ription of all the te hni al details is given

in this referen e. Here we shortly dis uss a few improvements of the HMC

algorithmthat are relevant fromthe pointof view of this thesis.

Forsomelatti eDira operators(e.g. Wilsontwistedmass),itispossible

to de ompose the Dira matrix intosubspa es of even and odd latti e sites,

thus redu ing the dimension of the problem. Su h te hnique is alled

even-odd pre onditioning [126℄.

Another approa h is to use more than one set of pseudo-fermion elds,

i.e. split the fermion determinant into two (or more) parts. One of the

widely used methods of this kind is alled the Hasenbus h tri k (or mass

pre onditioning)[127, 128℄and onsistsinutilizingtheidentity(examplefor

the

N f = 2

^ase ^with ^degenerate ^quark ^masses

µ

^):

| det( ˆ D)| ² = det( ˆ D ˆ D ^† + µ ² ) det D ˆ ˆ D ^† D ˆ ˆ D ^† + µ ²

!

.

^(C.1)

Su h de omposition splits the ontribution of the low-frequen y and

high-frequen y modes of

D ˆ ˆ D ^†

^and ^thus ^redu
es ^the ^ondition ^number ^of ^the

problem. It also allows for integration of dierent parts of the a tion on

dierenttime s ales, su hthat the most expensive part an besimulated on

the oarsest time s ale. A general version ofan HMC algorithm

in orporat-ing even-odd pre onditioning, mass pre onditioning and multiple time s ale

integration waspresented by Urba h,Jansen, Shindlerand Wenger[129℄. It

wasalsoshownby numeri alinvestigationthatsu hversionof thealgorithm

the smallquark mass limit. This version of the algorithmis the base of the

tmLQCD suite initspart that wasrelevant forthe generation ofgauge eld

ongurations used inthis proje t.

Anotherimportantimprovementof theHMC algorithm omesunder the

name ofPolynomialHMC (PHMC).It wasintrodu edand analyzedin[115,

116, 117, 118℄. This version of the algorithm an be applied to simulate

non-degenerate quarks. It is used e.g. in the tmLQCD suite in simulations

in ludingthe strangeand harm quark.

The number of other improvements of the HMC algorithm is very large

and isstillin reasing. Tonalizethis appendix wejustmention afewmore.

Fortheirdes ription werefertooriginalpapers. A wideand important lass

of improvements on ern integration s hemes and ome under the name of

multiple time-s ale integration. The generalization of the leap-frog s heme

to multiple time s ales was originally proposed by Sexton and Weingarten

[130℄. Another approa h is the so- alledse ond order minimalnorm (2MN)

integrator[131,132℄. Avariantof the HMCalgorithm alledRationalHMC

(RHMC) was dis ussed in [133, 134, 135, 136℄. Domain-de omposed HMC

was introdu ed in a series of papers by Lüs her [137, 138, 139℄ and later

augmented by low-mode deation [140℄.

Tree-level test of zero modes

subtra tion

In this appendix, we show the results of a free-eld test of routines used to

subtra t the zero modes (we will refer tothem as subtra tion routines) at

the level of propagators. The test is performed on a small latti e of

4 ³ × 8

with quarkmass set to

am = 0.2

^. ^W^e^perform ^thesubtra tion inthree ways for the pseudos alar (PP) and s alar (SS) orrelation fun tion, using:

•

^formula ^(2.16) ^for ^the ^PP ^orrelator ^and ^an ^analogous ^formula ^for ^SS

(analyti alformula),

•

^GWC ^ode ^withsubtra tion routines for pointsour es,

•

^GWC ^ode ^withsubtra tion routines for sto hasti sour es.

Usingnotation of Se tion 4.3, we write the mesoni orrelation fun tion

as:

C(t) = C ₀₀ (t) + 2C _0N (t) + C _{N N} (t),

^(D.1)

where the rst two terms involve the zero modes. Computing orrelation

fun tionsfromthefullpropagator(withallmodes)leadsto

C(t)

^,^while^if^the

zero modes are subtra ted atthe levelof propagators, onlythe part

C N N (t)

isobtainedbyperforming ontra tions,i.e. subtra tionofzeromodes an els

both the diagonal ontribution

C ₀₀ (t)

^and ^the ^mixed ^one

C _0N (t)

We remind here the formulafor the pseudos alar orrelator (2.16) and

gen-eralize itto in ludethe s alar ase:

C(t) = N _c N _d

where we obtain the pseudos alar orrelator by hoosing:

s(ξ) = 1

^for ^all

ξ

and the s alar orrelator if we take

s(ξ) = −1

^for

ξ = 0

^and

s(ξ) = 1

^for

ξ = 1, 2, 3, 4

To isolate the ontribution of the zero-modes, we have to al ulate the

diagonalpart

C ₀₀ (t)

^and ^the ^mixed ^part

C _0N (t)

latter orresponds to non-zero modes in the mixed term).

The ontribution of the zero modes is

C 00 (t) + 2C 0N (t)

^and ^it ^is ^for ^the

4 -0.046231202662437964612252 02

5 -0.048644806504103299538144 26

6 -0.055983137376237557258917 83

7 -0.068542693495896617195128 90

As we have shown analyti ally in Se tion 4.3, the ontribution of the zero

modes isthe sameinboththepseudos alarandthes alar orrelator,up toa

sign,whi hisamatterof onvention. Withsu h onvention,the ontribution

of the zero modes exa tly an les inthe sum

C _{P P +SS} = C _{P P} + C _SS

Wealsoshowthe part

C N N (t)

^of^the ^PP^and ^SS orrelators,i.e. the part with zero modes subtra ted:

t C_PP(t)

0 3.2070495598256036906548160 9

1 0.0538755232498468739077424 1

2 0.0118515945140288403436557 0

3 0.0043685036209793262629297 3

4 0.0025747032878168582548106 3

5 0.0043685036209795205519590 4

6 0.0118515945140288542214435 1

7 0.0538755232498471792190741 8

t C_SS(t)

0 -2.899846672457193630378924 35

1 0.0534957243448562630439013 8

2 0.0097986066820899761498076 0

3 0.0002349545374039987977177 2

4 -0.002407409019565424912467 93

5 0.0002349545374036310363408 1

6 0.0097986066820900039053832 1

7 0.0534957243448555622156170 8

Wewill use these numbers to ompare with the subtra tionroutines.

W dokumencie Uniwersytet im. Adama Mickiewicza Adam Mickiewicz University (Stron 148-154)

x

(S(~x, t;~0, 0) S † (~x, t;~0, 0)).

S(~x, t;~0, 0) = 1

N

N 4

C P P (t) = 1

δ(~p − ~p ′ ) = 1

S(p) =

4

5

X

ξ=0

S ξ (p)γ ξ ,

γ 0 ≡

ξ

C P P (t) = N c N d

N 3 N 4 2 X

~ p

X

p 4 ,p ′ 4 4

5

X

ξ=0

S ξ (~p, p 4 ) S ξ ∗ (~p, p ′ 4 ))e i(p 4 −p ′ 4 )t ,

N d =

(

)

N c =

(

c )

N f = 2

µ

| det( ˆ D)| 2 = det( ˆ D ˆ D † + µ 2 ) det D ˆ ˆ D † D ˆ ˆ D † + µ 2

!

.

D ˆ ˆ D †

4 3 × 8

am = 0.2

•

•

•

C(t) = C 00 (t) + 2C 0N (t) + C N N (t),

C(t)

C N N (t)

C 00 (t)

C 0N (t)

C(t) = N c N d

s(ξ) = 1

ξ

s(ξ) = −1

ξ = 0

s(ξ) = 1

ξ = 1, 2, 3, 4

C 00 (t)

C 0N (t)

C 00 (t) + 2C 0N (t)

C P P +SS = C P P + C SS

C N N (t)

(S(~x, t;~0, 0) S ^† (~x, t;~0, 0)).

δ(~p − ~p ^′ ) = 1

S _ξ (p)γ _ξ ,

N ³ N ₄ ² X

p 4 ,p ^′ ₄ 4

S ξ (~p, p 4 ) S _ξ ^∗ (~p, p ^′ ₄ ))e ^i(p ⁴ ^−p ^′ ⁴ ^)t ,

| det( ˆ D)| ² = det( ˆ D ˆ D ^† + µ ² ) det D ˆ ˆ D ^† D ˆ ˆ D ^† + µ ²

D ˆ ˆ D ^†

4 ³ × 8

C(t) = C ₀₀ (t) + 2C _0N (t) + C _{N N} (t),

C ₀₀ (t)

C _0N (t)

C(t) = N _c N _d

C ₀₀ (t)

C _0N (t)

C _{P P +SS} = C _{P P} + C _SS