Comparison of filtering methods in SU(3) lattice gauge theory

0.16 MB·English

by  F. Bruckmann

Comparison of filtering methods in SU(3) lattice
gauge theory
9 F. Bruckmanna, F. Gruber∗a, C. B. Langb, M. Limmerb, T. Maurera, A. Schäfera and S.
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2 aInstitutfürTheoretischePhysik,UniversitätRegensburg,
D-93040Regensburg,Germany
n
a bInstitutfürPhysik,FBTheoretischePhysik,Karl-Franzens-UniversitätGraz,
J
A-8010Graz,Austria

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E-mail: [email protected],

] [email protected],
t
a [email protected],
l
- [email protected],
p
[email protected],
e
h [email protected],
[ [email protected]

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Wesystematicallycomparefilteringmethodsusedtoextracttopologicalexcitationsfromlattice
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8 gaugeconfigurations.Weshowthatthereisastrongcorrelationofthetopologicalchargedensities
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obtainedbyAPEandStoutsmearing. Furthermore,afirstquantitativeanalysisofquenchedand
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. dynamicalconfigurationsrevealsacrucialdifferenceoftheirtopologicalstructure:thetopological
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0 chargedensityismorefragmented,whendynamicalquarksarepresent.Thisfactalsoimpliesthat
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smearinghastobehandledwithgreatcare,nottodestroythesecharacteristicstructures.
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8thConferenceQuarkConfinementandtheHadronSpectrum
September1-62008
Mainz,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Comparisonoffilteringmethods F.Gruber
1. Filteringmethods
ManymethodshavebeendevelopedtoextracttheIRcontentfromlatticedata. Unfortunately,
allthesemethodsintroduceambiguitiesandparameters. Thus,togetacoherentpictureofthetopo-
logicalstructureoftheQCDvacuum,itisnecessarytofindwaysofcontrollingorevenremoving
theseambiguities.
OneofthefirstattemptstofilterouttheUV“noise”hasbeenAPEsmearing[1],definedas:
(cid:110) α (cid:111)
UAPE=P (1−α )Uold+ APE(staples) , (1.1)
µ SU(Nc) APE µ 6
where α determines the weight of the old link and the sum of the attached staples. The right
APE
handsidehastobeprojectedbacktothegaugegroup. Unfortunately,thereisnouniquemapping.
One approach is to take the unitary part of the polar decomposition and normalize this matrix by
itsdeterminant. Stoutsmearing[2]circumventsthisprojectionbyusingtheexponentialmap:
(cid:110)i (cid:111)
UStout=exp Q (U,ρ ) ·Uold, (1.2)
µ 2 µ µν µ
whereQ (U,ρ )isahermitianmatrixconstructedfromallplaquettescontainingtheoldlinkU
µ µν µ
andweightedbyfactorsρ . Weusethecommonchoiceρ =ρ forisotropicsmearing.
µν µν Stout
ArelativelynewmethodisLaplacefiltering[3]. Thefilteredlinksareobtainedfromaspectral
sumofthelowesteigenmodesofthecovariantlatticeLaplacian1:
(cid:40) (cid:41)
N
ULaplace(x)=P −∑λ Φ (x)⊗Φ†(x+µˆ) . (1.3)
µ SU(Nc) n n n
n=1
This procedure acts as a low-pass filter in the sense of a Fourier decomposition. At this point it
shouldbestressedthatLaplacefilteringiscompletelydifferentfromsmearing,becauseitisbased
on rather global objects, namely the eigenmodes, and does not locally modify the gauge links in
contrasttosmearing.
Takingthefilteredlinksasastartingpoint,onecanreconstructthetopologicalchargedensity
q(x)=Tr(cid:0)Fµν(x)F(cid:101)µν(x)(cid:1)/16π2 fromanimprovedfieldstrengthtensor[4].
Also the fermionic definition of the topological charge, via the eigenmodes ψ of a chiral
Diracoperator,hasbeenusedtoexploretheIRstructure[5]. ForthissocalledDiracfilteringone
truncatesthesumin
N (cid:18)λ (cid:19)
q (x)= ∑ n −1 ψ†(x)γ ψ (x) (1.4)
Dirac 2 n 5 n
n=1
and takes only the lowest N modes into account. While the zero-modes determine the total topo-
logicalchargeQ=∑q(x)duetotheindextheorem,thenonzero-modesmodifythelocalstructure
ofthedensity,leavingthetotalchargeunaffected.
2. Comparisonofthedifferentmethods
In an earlier study a qualitative and quantitative similarity of the introduced filtering meth-
ods for quenched SU(2) gauge configurations has been observed [6]. One central element of this
1Theoriginallinkisreproducedforalleigenmodes,N=Nc·Vol,withnoprojectionneeded.
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Comparisonoffilteringmethods F.Gruber
comparisonisthecorrelatoroftwotopologicalchargedensitiesq (x)andq (x)definedby:
A B
(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)
χ ≡ 1/V ∑ q (x)−q q (x)−q , (2.1)
AB A A B B
x
wherethemeanvaluesaresubtractedforconvenience. Fromthiswecanconstructaquantitythat
reflectsthe“matching”oftwomethods:
χ2
Ξ ≡ AB (2.2)
AB
χ χ
AA BB
Ξ isobviouslyequaltoone,ifq (x)isproportionaltoq (x)anddeviatesthemorefromone,the
AB A B
morethedensitiesdiffer.
The main idea is now to relate different filter parameters for those combinations where Ξ is
maximal. Infig.1thecontourlinesofΞforseveralmethodsandparameterrangesareshown. On
therighthandplottwoexemplarycombinationsareindicatedthatcorrespondtothebestmatching
valuefordifferentfilteringstrengths.
Aninterestingobservationisthatthereisanalmostone-to-onecorrespondencefornstepsof
APEandnstepsofStoutsmearingwhenα ≈6·ρ . Asseenintheplotonthelhs.offig.1,
APE Stout
Ξ>0.95foralargenumberofsmearingsteps. ThisisconsistentwithresultsbyCapitanietal.[7],
wheresucharelationhasbeenderivedfromperturbationtheory. Whiletheyhavefocusedonglobal
observables with up to 3 smearing steps, our nonperturbative result reflects the local similarity of
bothmethodsandtheirstronglycorrelatedtopologicalchargedensitiesupto50steps.
50
40
5)
07 Ξ>0.95 2 L
=0. 30 4
ρ
s ( 6
p 10
e
st 20 0.495
ut 20
o
St 40
10
0.739
80
160
0
0 10 20 30 40 50 S1 2 10 2030 50 80 160
APE steps (α=0.45)
Figure1: LevelcurvesofΞ=0.95,0.85,...(startingfromthediagonal)forAPEvs. Stoutsmearing(left)
andΞ=0.8,0.7,...(startingfromtheinside)forAPEsmearing(S)vs. Laplacemodes(L)(right). (cid:78)and•
marktwoexamplesof“matching”parametersforweakandstrongfilteringrespectively(from[6]).
3. Clusteranalysisofthetopologicalchargedensity
Another important challenge is to extract observables from lattice data, that could be com-
paredwithcontinuummodelsofthevacuum. Onepossibilityistoanalyzetheclusterstructureof
the topological charge density. Two lattice points belong to the same cluster, if they are nearest
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Comparisonoffilteringmethods F.Gruber
dynamical dynamical
0.9 quenched quenched
0.8 n
1000 103um
nt be
pone 0.7 instanton gas, Nf=2 r of c
ex 0.6 instanton gas, quen. lus
te
rs
0.5 100 102
0.4
6 7 8 910 15 22 30 40 6 7 8 910 15 22 30 40
smearing steps smearing steps
Figure2: left: Exponentξ oftheanalysisforclusterscommontoAPEandStoutsmearing. Thesolidlines
showthevaluespredictedfromthediluteinstantongas.right:Totalnumberofdistinctclustersforaconstant
fraction f =0.0755ofpointslyingabovethecut-off. Lessthan6stepsarenotconsidered,asthedefinition
of the topological charge gets ill-defined. Errors have been calculated using an ensemble average over 10
configurationsbutarepartlytoosmalltosee.
neighbors and have the same sign of the topological charge density. Bruckmann et al. [6] found
a power law for the number of clusters as function of the ratio of points with |q(x)| lying above
a variable cut-off q and the total number of lattice points. The exponent ξ of this power law is
cut
highly characteristic for the topological structure of the QCD vacuum. Different models lead to
differentpredictions, whichallowsforaverysensitivetest. Ifonehasforinstancepurenoise, the
exponent is 1, as every point forms its own cluster. On the other hand one will have an exponent
closetozeroforverysmoothdensitieswithlargestructures.
Toreduceambiguitieswetakeonlythoseclustersintoaccount,whicharecommontodifferent
filters,whoseparameterswerematchedaccordingtomaximalvaluesofΞ. So,ifthereisanartifact
comingfromonemethod,itisunlikelythatthisartifactwillalsobeseenbytheother.
TheexponentforclusterscommontoAPEandStoutcanbefoundinfig.2(left). Weusedone
quenched and one dynamical N =2 ensemble with equal lattice spacing (see tab. 1). Obviously
f
theexponentsofthedynamicalconfigurationslieabovethequenchedvalues.
In order to interpret the cluster exponent, a model of dilute quantized topological objects of
general shape and with a size distribution d(ρ)∼ρβ has been considered in [6]. It leads to ξ =
(1+4/(β+1))−1 (in4dimensions). Followingthismodel,ourfindingsgivealargercoefficientβ
inthedynamicalcase. Hence,smallertopologicalobjectsbecomesuppressed.
Moreover, the rhs. of fig. 2 shows that for a fixed number of points, lying above the cut-off,
much more clusters are found in the dynamical case. Thus we conclude that when fermion loops
are taken into account, the topological structure is more complex and fragmented, in the sense of
larger number of distinct objects per volume. This seems to be in accordance to the findings of
the Adelaide group, where small instantons have been seen to be suppressed in the presence of
dynamicalquarks,whilethetotalnumberofinstantonsincreased,seefig.6in[8].
Thedifferenceoftheclusterexponentsquenchedvs.dynamicalvanishesforstrongersmearing
(∼30steps)andtheexponentssettledowntothesameplateau. Sowehavereasonstobelievethat
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Comparisonoffilteringmethods F.Gruber
toomuchsmearingdestroystheimpactofdynamicalquarks.
On the lhs. of fig. 2 we have included for comparison the exponents ξ = 7/11 ≈ 0.64 and
ξ =23/35≈0.66 for the SU(3) instanton gas without resp. with dynamical quarks. Taking the
dilute instantongas as asimplified model, itis obvious thatthe true vacuumshould have ahigher
exponent,asmorestructuresarepresent. However,theresultinfig.2(left)showsthatthisisonly
the case for very few smearing steps, for slightly stronger filtering we reach smoother configura-
tionsthanpredictedbythediluteinstantongas. Thisisanotherindicationofsmearingartefacts.
lat. size lat. spacing β m
LW 0
quenched 163·32 0.148 7.90 –
dynamical 163·32 0.150 4.65 -0.060
Table 1: Ensembles were generated with the Lüscher-Weisz gauge action and a chirally improved Dirac
operator[9]. Forthedynamicalsimulationstwoflavorsofmassdegeneratelightquarkswereused[10].
4. Conclusionandoutlook
Inconclusion,wehavefoundastrongcorrelationofthetopologicalchargedensitiesobtained
from APE and Stout smearing. Furthermore, our first results for dynamical quarks imply that the
topologicalstructureismorecomplexandfragmentedinthepresenceoffermionloops. Butthere
are also indications that smearing has to be used with great caution, especially when dealing with
dynamical configurations. The smallness of the cluster exponent is a sign that smearing is more
destructiveinSU(3)thaninSU(2). PreliminaryresultsforclusterscommontoAPEsmearingand
Laplace filtering do not show such artefacts, as Laplace filtering preserves smaller objects better.
Thiseffectisunderinvestigation.
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