Comparative study of topological charge

9.3 MB·English

by  Yusuke Namekawa

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Comparative study of topological charge
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t YusukeNamekawa∗
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l CenterforComputationalSciences,UniversityofTsukuba,Tsukuba,Ibaraki305-8577,Japan
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p E-mail: [email protected]
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Comparativestudy of topologicalcharge is performed. Topologicalchargesare measured by a

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cloverleaf operator on smoothed gauge configurations. Various types of smoothing techniques
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ined.Highconsistencyisobservedbetweentopologicalchargesobtainedbyimprovedsmoothing
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The32ndInternationalSymposiumonLatticeFieldTheory,
23-28June,2014
ColumbiaUniversityNewYork,NY
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Comparativestudyoftopologicalcharge YusukeNamekawa
1. Introduction
A topological charge is one of the most fundamental quantity in QCD. It characterizes the
vacuum structure. Lattice QCD is a main tool in the study of the topological charge [1]. Lattice
QCDallowsustoperformanonperturbative analysis inasystematic way.
Thetopological charge isoften measured withagluonic fieldstrength operator onthelattice.
Though it suffers from noisy ultraviolet fluctuations, a smoothing technique tames them so that a
discernible signal of the topological charge can be obtained. Cooling or smearing have been used
forsmoothinggaugefields. Recently,agradientflowisalsoemployed. Incontrasttothetraditional
cooling and smearing, the gradient flow has an advantage that it provides a continuous change of
thegaugefield. Thegradient flowaccomplishes abettercontrol ofsmoothing.
Alternatively, the topological charge can be calculated with afermionic definition. Thetopo-
logicalchargeisdetermined,forexample,bytheindextheoremwiththeoverlap-Diracoperator. A
clearadvantageofthefermionicdefinitionisthattheresultisguaranteedtobeaninteger. Asubtle
point, ontheotherhand,istheintegervaluesdependonthechoiceofthedefinition, duetoafinite
lattice spacing. In the case of the overlap-Dirac operator, a value of the topological charge occa-
sionallychangesaccordingtoaparameterintheformulation. Consistencycheckofthetopological
charges in the fermionic and gluonic definitions would be helpful as an estimator of the scaling
violation.
In this work, topological charges are computed with gluonic operators on N = 2 topology
f
fixedgaugeconfigurations. Themeasurements areperformedusingseveralsmoothing techniques.
Cooling with plaquette and improved local actions, APE and HYP smearing, as well as gradient
flowsareemployed. Theresults arecompared witheachother, andwiththevalues obtained using
theoverlap-Dirac operator. Similarattemptsarereported inRefs.[2].
2. Setup
2.1 Gaugeconfiguration
MeasurementofthetopologicalchargeisperformedonN =2gaugeconfigurationsprovided
f
byJLQCDCollaboration [3]. Thelatticesizeis163×32atthelatticespacingofa=0.118(2)fm.
ThegluonactionisIwasaki-type improvedgaugeaction,
Sg=b cg0 (cid:229) Pmn (x)+cg1 (cid:229) Rmn (x) , (2.1)
x,m <n x,m ,n !
whereb =6/g2,cg=3.648, cg=−0.331. Thequarkactionisanoverlap-Dirac fermionaction,
0 0 1
S = q¯D (m)q, (2.2)
q ov
m m
D (m) = m + + m − g sgn(H (−m )),H (−m )=g D (−m ), (2.3)
ov 0 0 5 W 0 W 0 5 W 0
2 2
(cid:16) (cid:17) (cid:16) (cid:17)
where m is the bare quark mass. D (−m ) is the Wilson operator with a negative mass, −m =
W 0 0
−1.6. Furthermore, unphysical Wilson fermion y with a negative mass as well as twisted mass
termsareaddedtofixthetopological chargedefinedbytheindextheorem Q ,
index
d S =y¯D (−m )y +f †(D (−m )+img t )f , (2.4)
W W 0 W 0 5 3
2
Comparativestudyoftopologicalcharge YusukeNamekawa
b m Q #conf MDtime
index
2.30 0.05 -2 50 2500
Table 1: Parameters of the gaugeconfigurations. Molecular Dynamics time is the numberof trajectories
multipliedbythetrajectorylength.
where f is a pseudofermion, and m is the twisted mass parameter. m = 0.2 is employed in the
configuration generation. Each configuration is separated by 100 trajectories with its trajectory
length0.5. Themainsimulationparameters aresummarizedinTable1.
2.2 Gluonictopological chargeoperator
Thetopological chargeismeasured usingagluonic fieldstrength.
Q = c Q1×1+c Q1×2, (2.5)
improve 0 1
Q1×1,2 = 321p 2 (cid:229) e mnrs TrFmn1×1,2Frs1×1,2, (2.6)
m ,n ,r ,s
i 1
Fmn1×1,2 = − [Cm1n×1,2]AH,CmAn H = Cmn −Cm†n , (2.7)
4 2i
(cid:0) (cid:1)
where C1×1 is the cloverleaf constructed with 1×1 plaquettes, and C1×2 with 1×2 rectangular
mn mn
loops. The improvement coefficients c and c can be tuned to reduce the scaling violation in the
0 1
topological charge operator. Threetypesof(c ,c )areinvestigated.
0 1
(c ,c ) = (1,0) Naive-type, (2.8)
0 1
= (5/3,−1/12) Symanzik-type, (2.9)
= (3.648,−0.331) Iwasaki-type. (2.10)
Figure 1 displays c dependence of the topological charge on a single configuration smoothed by
1
Wilson flow. The topological charge with Symanzik-type coefficients has the smallest deviation
from an integer. Since the deviation is originated from the finite lattice spacing, it implies an
efficient reduction of the scaling violation by Symanzik-type operator. Calculations on other con-
figurationsshowasimilartendency. Basedonthisresult,Symanzik-typecoefficientsareemployed
inthiswork.
2.3 Smoothing
Three kinds of smoothing techniques are evaluated: cooling, smearing, and gradient flow.
Smoothingisrequiredtosuppressnoisyultraviolet fluctuations, whilekeepingatopological struc-
ture. Although any smoothing is expected to give a consistent result in the continuum limit, it is
valuable tofindamethodthathastheleastlatticeartifact.
Coolingeliminatesultraviolet noisesbyreplacing eachlinkvariablesuchthatthelocalaction
is minimized [4]. For the local action, not only a naive plaquette action, but also Symanzik and
IwasakiactionsareemployedwiththecoefficientsofEq.(2.8)–(2.10).
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Comparativestudyoftopologicalcharge YusukeNamekawa
0.20
163 · 32
e)| b = 2.30, mud = 0.050
v Qindex = -2
o 0.15
pr Wilson flow, t = 200
m Ntraj = 50
Qi
nt( 0.10
e - i
v Iwasaki
o
mpr 0.05 Naive
Qi Symanzik
|
0.00
-0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05
c1
Figure1:Improvementcoefficientc dependenceofimprovedtopologicalchargeQ onasinglegauge
1 improve
configuration.DifferenceofQ fromanintegerisplotted.
improve
Anotherwayofsmoothing issmearing. APEsmearing[5]isdefinedby
a
Umnew(x) = ProjSU(3) (1−a )Um (x)+ 6S m (x) ,a =0.6. (2.11)
S m (x) = (cid:229) Un (hx)Um (x+n )Un†(x+m ) i (2.12)
±n 6=m
Inaddition, HYPsmearing[6]isalsoexamined.
a
Umnew(x) = ProjSU(3) (1−a 1)Um (x)+ 61 (cid:229) Un(2;m)(x)Um(2;n)(x+n )Un(2,m),†(x+m ) , (2.13)
" ±n 6=m #
a
Um(2;n)(x) = ProjSU(3) (1−a 2)Um (x)+ 42 (cid:229) Ur(3;mn) (x)Um(3;n)r (x+r )Ur(3,mn),†(x+m ) (,2.14)
" ±r 6=m ,n #
a
Um(3;n)r (x) = ProjSU(3) (1−a 3)Um (x)+ 23 (cid:229) Us (x)Um (x+s )Us†(x+m ) , (2.15)
" ±s 6=m ,n ,r #
a = 0.75,a =0.6,a =0.3. (2.16)
1 2 3
Smearedgaugefieldsareprojected backtoSU(3)byMaximumSU(3)projection.
ProjmaxSU(3)(Um (x)) = max ReTr(Umnew(x)Um†(x)). (2.17)
SU(3) Umnew(x)∈SU(3)
Analternative smoothing isgivenbythegradient flow[7]. Theevolution ofthegaugefieldis
determined by
¶ S
¶ tVm (x,t) = −Vm (x,t)¶ Vm (x), (2.18)
Vm (x,t =0) = Um (x), (2.19)
where t is the flow time, and S is an action without its coupling constant. Similar to the cooling
case, plaquette, Symanzik, and Iwasaki actions are employed. The flow equation is solved by the
fourth order Runge-Kutta in the commutator-free method [8]. The Runge-Kutta step size dt is
chosentobe0.02. Thesystematicerrorassociated withdiscretization oftheflowtimeisdefinitely
belowthestatistical error.
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Comparativestudyoftopologicalcharge YusukeNamekawa
2.0 163 · 32 Cooling(plaq) 2.0 163 · 32 Cooling(Symanzik) 2.0 163 · 32 Cooling(Iwasaki)
1.0 b = 2.30, mud = 0.050 1.0 b = 2.30, mud = 0.050 1.0 b = 2.30, mud = 0.050
Qindex = -2 Qindex = -2 Qindex = -2
0.0 0.0 0.0
mprove-1.0 mprove-1.0 mprove-1.0
Qi Qi Qi
-2.0 -2.0 -2.0
-3.0 -3.0 -3.0
-4.00 50 100 150 200 250 300 -4.00 50 100 150 200 250 300 -4.00 50 100 150 200 250 300
Cooling step Cooling step Cooling step
2.0 163 · 32 APE 2.0 163 · 32 HYP
1.0 b = 2.30, mud = 0.050 1.0 b = 2.30, mud = 0.050
Qindex = -2 Qindex = -2
0.0 0.0
mprove-1.0 mprove-1.0
Qi Qi
-2.0 -2.0
-3.0 -3.0
-4.00 100 200 300 400 500 600 -4.00 100 200 300 400 500 600
Smearing step Smearing step
2.0 163 · 32 Wilson flow 2.0 163 · 32 Symanzik flow 2.0 163 · 32 Iwasaki flow
1.0 b = 2.30, mud = 0.050 1.0 b = 2.30, mud = 0.050 1.0 b = 2.30, mud = 0.050
Qindex = -2 Qindex = -2 Qindex = -2
0.0 0.0 0.0
mprove-1.0 mprove-1.0 mprove-1.0
Qi Qi Qi
-2.0 -2.0 -2.0
-3.0 -3.0 -3.0
-4.00 50 100 150 200 -4.00 50 100 150 200 -4.00 50 100 150 200
3 · flow time 3 · flow time 3 · flow time
Figure2: SmoothingstepdependenceofQ usingcooling(toppanels),smearing(middlepanels),and
improve
gradientflow(bottompanels).
2.4 Results
Figure2illustratescoolingandsmearingstepdependenceoftheimprovedtopologicalcharge.
The flow time dependence is also plotted. The flow time is multiplied by a factor of three, which
is expected from a perturbative analysis [9]. In every case, the topological charge has an integer
value with a sufficiently large number of steps. A small number of smoothing steps leads to a
fake plateau i.e. a semi-stable value of the topological charge. The number of smoothing steps is
determinedtosatisfytheadmissibilitycondition, max[ReTr(1−U )]<0.067[10,11]. InFig.3,
plaq
Wilsonflowtimedependenceoftheplaquetteisshown. Nojumpofthetopologicalchargeseemto
betriggered,iftheadmissibilityconditionisfulfilled. Itshouldbementionedmax[ReTr(1−U )]
plaq
doesnotalwaysdecreaseastheflowtimegrows,thoughthevaluesummedoverthespacetimefalls
offmonotonically.
Figure4presentshistogramsoftheimprovedtopologicalcharges. Sincethetopologicalcharge
determinedbytheindexQ isfixedintheconfigurationgenerations,thehistogramisexpectedto
index
haveasharppeakaroundQ ,supposingthescalingviolationissmall. Thehistogramsobtained
index
by cooling withimproved local actions show the expected behavior. Almostall ofthe topological
charges agree with Q . On the other hand, cooling using the plaquette action has a broad his-
index
togram. It implies a relatively large lattice artifact in the unimproved cooling method. Analogous
trends are observed inother smoothing procedures. HYPsmearing has anarrow histogram, while
APEsmearing does not. Symanzik and Iwasaki flowsform asharp peak inthe histogram. Onthe
contrary, Wilson flow brings a wide peak. Improved smoothing methods leads to higher consis-
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Comparativestudyoftopologicalcharge YusukeNamekawa
100 100
163 · 32 Wilson flow 163 · 32 Wilson flow
10 b = 2.30, mud = 0.050 10 b = 2.30, mud = 0.050
Qindex = -2 Qindex = -2
q) 1 1
a
1.0 - pl 00.0.11 0 - plaq 00.0.11
x( 1.
a
m
0.001 0.001
0.0001 0.0001
1e-05 1e-05
0 50 100 150 200 0 50 100 150 200
3 · flow time 3 · flow time
Figure3: Wilsonflowtimedependenceofmaximumof(1−plaquette)(leftpanel),andthevaluesummed
overthespacetimevolume(rightpanel).
tency with Q , indicating the scaling violation in the topological charge is suppressed well by
index
theimprovement.
2.5 Conclusion
Systematiccomparisonoftopologicalchargesispresented. Topologicalchargesaremeasured
on N = 2 topology fixed configurations. Several smoothing techniques are evaluated using a
f
gluonictopologicalchargeoperatorofSymanzik-typecoefficients,whichgiveatopologicalcharge
withthesmallestdeviation fromaninteger.
Cooling with improved actions, HYPsmearing, and improved gradient flows are found to be
advantageous. Morethan90%ofthetopological charges areconsistent withthoseobtained bythe
indextheorem. Itindicates theirlatticeartifactsarereducedefficiently. Ontheotherhand,cooling
with plaquette action, APE smearing, and Wilson flow lead to partial matches. The agreement is
limitedto70-80%. Scalingviolations seemtobecomparatively largeinthesesmoothingmethods.
Scaling properties as well as finite size effects of the topological charge have not been inves-
tigated. It is important to estimate them, but is beyond the scope of this work due to limitation of
thegaugeconfigurations. Thegaugeconfigurations havebeengenerated atasinglelatticespacing
andspatial volume. Theseevaluations areleftforthefuturework.
Acknowledgments
I would like to thank members of MEXT SPIRE Field 5 Project 1 and Bridge++ develop-
ment team, as well as H.Fukaya and K-I.Nagai for valuable discussions. I am grateful to JLQCD
Collaboration forprovidingtheirgaugeconfigurations [3]. ThisworkisinpartbasedonBridge++
code[12,13,14]. ThisworkissupportedbyJLDGconstructedoverSINET3ofNII,MEXTSPIRE
andJICFuS,andGrants-in-AidforScientificResearchGrantNumber24540250.
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Comparativestudyoftopologicalcharge YusukeNamekawa
50 163 · 32 Cooling(plaq) 50 163 · 32 Cooling(Symanzik) 50 163 · 32 Cooling(Iwasaki)
45 b = 2.30, mud = 0.050 45 b = 2.30, mud = 0.050 45 b = 2.30, mud = 0.050
40 Qindex = -2 40 Qindex = -2 40 Qindex = -2
35 35 35
30 30 30
25 25 25
20 20 20
15 15 15
10 10 10
5 5 5
0-6 -5 -4 -3 -2 -1 0 1 2 0-6 -5 -4 -3 -2 -1 0 1 2 0-6 -5 -4 -3 -2 -1 0 1 2
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Qimprove Qimprove
50 163 · 32 Wilson flow 50 163 · 32 Symanzik flow 50 163 · 32 Iwasaki flow
45 b = 2.30, mud = 0.050 45 b = 2.30, mud = 0.050 45 b = 2.30, mud = 0.050
40 Qindex = -2 40 Qindex = -2 40 Qindex = -2
35 35 35
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5 5 5
0-6 -5 -4 -3 -2 -1 0 1 2 0-6 -5 -4 -3 -2 -1 0 1 2 0-6 -5 -4 -3 -2 -1 0 1 2
Qimprove Qimprove Qimprove
Figure 4: Histograms of Q with cooling (top panels), smearing (middle panels), and gradient flow
improve
(bottompanels).
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