Stochastic calculation of the QCD Dirac operator spectrum with Mobius domain-wall fermion

0.11 MB·

by  G. Cossu

KEK-CP-339,OH-HET-887
Stochastic calculation of the QCD Dirac operator
spectrum with Mobius domain-wall fermion
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1
0
2

n
a G. Cossu
J

5 HighEnergyAcceleratorResearchOrganization(KEK),Tsukuba305-0801,Japan

] H. Fukaya
t
a
l DepartmentofPhysics,OsakaUniversity,Toyonaka560-0043,Japan
-
p
S. Hashimoto∗
e
h HighEnergyAcceleratorResearchOrganization(KEK),Tsukuba305-0801,Japanand
[
SchoolofHighEnergyAcceleratorScience,SOKENDAI(TheGraduateUniversityforAdvanced

1 Studies),Tsukuba305-0801,Japan
v E-mail: [email protected]
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4
T.Kaneko
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HighEnergyAcceleratorResearchOrganization(KEK),Tsukuba305-0801,Japanand
0
0 SOKENDAI(TheGraduateUniversityforAdvancedStudies),Tsukuba305-0801,Japan
.
1
J. Noaki
0
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HighEnergyAcceleratorResearchOrganization(KEK),Tsukuba305-0801,Japan
1
:
v
i WecalculatethespectralfunctionoftheQCDDiracoperatorusingthefour-dimensionaleffective
X
operator constructed from the Mobius domain-wall implementation. We utilize the eigenvalue
r
a filteringtechniquecombinedwiththestochasticestimateofthemodenumber. Thespectrumin
theentireeigenvaluerangeisobtainedwithasinglesetofmeasurements. Resultson2+1-flavor
ensembleswithMobiusdomain-wallseaquarksatlatticespacing∼0.08fmareshown.
The33rdInternationalSymposiumonLatticeFieldTheory
14-18July2015
KobeInternationalConferenceCenter,Kobe,Japan*
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
StochasticcalculationofQCDDiracspectrum S.Hashimoto
1. Introduction
Theeigenvalue density(orspectralfunction) oftheDiracoperator
r (l )= 1h(cid:229) d (l −l )i (1.1)
i
V
i
provides aprobe ofthe spontaneous chiral symmetry breaking inQCDthrough the Banks-Casher
relationr (0)=S /p [1],whereS denotesthechiralcondensateS =−hq¯qiinthethermodynamical
limit. Thefunctional form ofr (l ) atsmall l is computed by one-loop chiral perturbation theory
in the p-regime [2] and in the mixed regime [3], but the value of S is to be determined by non-
perturbative QCDcalculations.
The most direct way of obtaining the eigenvalue density in lattice QCD is to calculate the
individual low-lying eigenvalues and to count the number of them falling in a region suffciently
close to zero. This method was adopted in our previous works to extract the chiral condensate in
2+1-flavor QCD[4,5]using the overlap-Dirac operator. Forlarger volumes, however, itbecomes
computationallymoredemandingbecauseofthecostandmemoryrequirementoftheLanczos-type
algorithms.
Analternativewayistostochasiticallyestimatethenumberofeigenvaluesbelowsomethresh-
old. Itwasfirstimplementedin[6]forthisparticular problem. Inthisworkweintroduce avariant
of this method tocalculate the spectral function. Namely, weutilize the Chebyshev filtering tech-
nique combined with a stochastic estimate of the mode number. As described in the next section,
the method is more flexible and can be used to calculate the whole spectrum at once. We use
the lattice ensembles generated with 2+1 flavors of the Mobius domain-wall fermion at a lattice
spacing a≃0.08fm.
2. Chebyshev filetering
Onecanevaluatethenumberofeigenvaluesinaninterval[a,b]ofahermitianmatrixA,which
issupposed tobeD†DofanylatticeDiracoperator D,as
n[a,b]= 1 (cid:229)Nv x †h(A)x (2.1)
N k k
vk=1
with Gaussian random vectors x , which hasanormalization (1/N )(cid:229) Nv x †x =12V in thelimit
k v k=1 k k
of large N , the number of random vectors. h(A) is a function of matrix A that works as a filter
v
of eigenvalues. Without h(A), (2.1) simply counts the total mode number. By preparing h(x)
returning 1 in the range [a,b] and 0 elsewhere, we may stochastically count the number of modes
in that interval. The statistical error is given by a square-root of the mode number in[a,b]. When
thenumberofeigenvaluesintherange[a,b]issufficientlylarge,N =1couldalreadygiveaprecise
v
estimate.
OnecanusetheChebyshevpolynomial T (x)toapproximate thefilterh(x):
j
p
h(x)= (cid:229) gpg T (x). (2.2)
j j j
j=0
2
StochasticcalculationofQCDDiracspectrum S.Hashimoto
1
0.8
0.6
0.4
0.2
0
0 0.005 0.01 0.015 0.02
al
Figure 1: Step function approximation given by the Chebyshev polynomial at order p=8000. Typical
resultsfortheinterval[0,d ]=0.01,0.005,0.002and0.001areshownfromrighttoleft.
The coefficients g and gp are known numbers fixed once the interval [a,b] is given. The conven-
j j
tionalChebyshevminmaxapproximationisobtainedwithg ,whiletheJacksonstabilizationfactor
j
p
g isintroducedtosuppresstheoscillationtypicalintheChebyshevexpansion[7]. Inorderforthe
j
Chebyshev approximation towork,thewholeeigenvalues ofAhavetobeintherange[−1,1].
Aftertheensembleaverage(overgaugeconfigurations) oneobtains
n¯[a,b]= 1 (cid:229)Nv (cid:229) p gpg hx †T (A)x i . (2.3)
N j j k j k
vk=1"j=0 #
An important observation is that once the stochastic estimates of hx †T (A)x i are calculated for
k j k
each j the eigenvalue count in any interval [a,b] can be obtained by combining them with the
corresponding coefficients gpg . Namely, the interval can be adjusted afterwards, independent of
j j
thecostlycalculation ofthepolynomial ofthematrixA. Detailsofthemethodarefoundin[7].
TheChebyshevpolynomial canbeeasilyconstructed usingtherecurrence formula
T (x)=1, T (x)=x, T (x)=2xT (x)−T (x). (2.4)
0 1 j j−1 j−2
One can also use the relations T (x)=2T (x)T (x)−T (x) and T (x)=2T2(x)−T (x), in
2n−1 n−1 n 1 2n n 0
order to reduce the numerical efforts. With infinitely large p the filtering function h(x) is exactly
reproduced; atfinite p,theapproximatingfunctionissmearedaroundthebordersaandb,inducing
asystematicerror.
Forthe four-dimensional effective Dirac operator ofdomain-wall fermion, the eigenvalues of
D†D are in the range [0,1]. In Figure 1, we plot the step functions for the interval [0,d ] of the
eigenvalue of |D| with d = 0.01, 0.005, 0.002 and 0.001. (For the correspondence between the
eigenvalues of D†Dand those of|D|, see below.) Thepolynomial order isfixedto p=8000. One
can see that the step function is well approximated away from the boundary. Near the boundary,
the edge is rounded off. Its effect is relatively more important for smaller d . The error estimated
forthearea,whichhastobed ,is0.8%ford =0.01and1.5%ford =0.005,scalingas1/d .
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StochasticcalculationofQCDDiracspectrum S.Hashimoto
3. Lattice calculation
The JLQCD collaboration has generated a new set of ensembles of 2+1-flavor QCD with
Mobius domain-wall fermion for sea quarks. It aims at achieving good chiral symmetry, i.e. the
residual massisorder1MeVorsmaller. Threelatticespacings arechosenas1/a=2.45,3.61and
4.50 GeV, which allow well controlled continuum extrapolation even including charm quarks as
valence quarks. Theup anddownquark masses correspond tothe pion massMp of230, 300, 400
and 500 MeV; two strange quark masses are chosen such that they sandwich the physical value.
Lattice volume is 323×64, 483×96 and 643×128, depending on the lattice spacing, and the
physical volume satisfies the nominal condition Mp L&4. This set of ensembles have been used
foravarietyofapplications [8,9,10,11,12].
In this preliminary work, we use the coarse lattice (1/a = 2.45 GeV) of size 323×64, out of
the abovementioned ensembles. Thenumber ofconfigurations is50foreach ensemble, taken out
of10,000HMCtrajectories. ThenumberN oftheGaussiannoisevectorx is1. Theupanddown
v k
quarkmassesinthelatticeunitaream =0.019,0.012,0.007and0.0035.
ud
We calculate the eigenvalue density of the hermitian operator D(4)†D(4) made of the four-
dimensional (4D)effectiveoperator
D(4)=[P−1(D(5)(m=1))−1D(5)(m=0)P] . (3.1)
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Here,D(5)(m)representsthefive-dimensional(5D)Mobiusdomain-walloperatorwithmassm. For
the eigenvalue count we took m=0, i.e. the massless Dirac operator. The 4D effective operator
is constructed by multiplying the inverse of the Pauli-Villas operator (m=1) and taking the 4D
surfaces (represented by the subscript “11”) appropriately projected onto left- and right-handed
modesbyaprojection operatorP. See,forinstance, [13]formoredetails.
For each application of D(4) on 4D vectors, we have to calculate the inverse of the Pauli-
Villarsoperator,forwhichtheconjugategradientiterationoforder40–50isinvolved. Althoughthe
inversion ismuchlessexpensivethanthecalculation oflightquarkpropagator, thetotalnumerical
costissubstantial becausewehavetomultiplyD(4)†D(4) p-times. (p=8000inthisanalysis.)
The eigenvalues of D(4)†D(4) are in the region [0,1], and we rescale the operator as A =
2D(4)†D(4)−1 to match the region of the Chebyshev approximation. Since the effective 4D op-
erator satisfies the Gisparg-Wilson relation very precisely, we assume that eigenvalues of D(4) lie
on a circle inthe complex plane. Inthe following, the eigenvalue l stands for that projected onto
theimaginary axisasl = l /(1−l ).
D(4)†D(4) D(4)†D(4)
Figure2showstheeigenvaluespectrum forthewholerangeofl inthelatticeunit. Bothaxes
p
are in a logarithmic scale. For each bin of [a,b], it is constructed as r (l ;d )=(1/2V)n¯[a,b]/d
withabinsized . (Therefore, itsatisfiesl = a/(1−a)andl +d = b/(1−b).)
One can clearly see that the number of eigenvalues increases toward higher l and saturate at
p p
some point of O(1) due to the discretization effect, which should otherwise behave like ∼l 3 for
asymptotically large l . There is no visible quark mass dependence in this region. On the lowest
end,itapproachesaconstantcorrespondingtor (0),fromwhichoneextractsthechiralcondensate.
Thesame data are plotted in Figure 3 inalinear scale. The individual bin has awidth of d =
0.005. With this binsize, the systematic error due to the Chebyshev approximation is well below
thestatistical error.
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StochasticcalculationofQCDDiracspectrum S.Hashimoto
1
am = 0.019
ud
0.1 am = 0.012
ud
am = 0.007
ud
) am = 0.0035
l( 0.01 ud
3r
a
0.001
0.0001
0.001 0.01 0.1 1 10
al
Figure 2: Spectral function in a logarithmic scale. The lattice data are plotted for four values of up and
downquarkmasses.
0.001
)
l( 0.0005
3r
a m = 0.019, m = 0.040
ud s
m = 0.012, m = 0.040
ud s
m = 0.007, m = 0.040
ud s
m = 0.0035, m = 0.040
ud s
0
0 0.02 0.04 0.06 0.08 0.1
al
Figure3: Spectralfunctioninalinearscale. Thebinsizeistakenasd =0.005. Thelatticedataareplotted
for four values of up and down quark masses (histogram with four differentcolors). Curves are those of
one-loopchiralperturbationtheory.
4. Analysisusing c PT formula
Figure3showsacleardependence ofr (0)ontheseaupanddownquarkmassm . Namely,
ud
r (0)gets lowerforsmaller m . Furthermore, apeak develops near l =0for heavier seaquarks.
ud
Qualitatively, it is understood as the effect that the fermion determinant is no longer active below
l .m tosuppress thenear-zero modes. Morenear-zero eigenvalues maythensurvive forlarger
ud
m .
ud
In order to obtain the chiral condensate S , one has to take the thermodynamical limit, i.e.
the infinite volume limit and then the massless quark limit. The order of the limits is crucial;
r (0) vanishes in the massless limit on any finite volumes. Fortunately, such volume and mass
dependences are well understood in chiral perturbation theory (c PT), and we may identify the
volume beyond which the system is effectively in the large volume limit. All our lattices satisfy
thatcondition, andweusethe p-regime c PTformulainthefollowinganalysis.
One-loop formula for N = 2 is available in [2]. It is written in terms of the leading order
f
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StochasticcalculationofQCDDiracspectrum S.Hashimoto
0.0007
0.0006
]
1 0.0005
0
0.
0,
[
0.0004
3r
a
0.0003
0.0002
0 0.01 0.02
am
ud
Figure4: Chiralextrapolationofthepartiallyintegratedspectralfunctionr [0,d ]withd =0.01. Theone-
loopc PTcurveatN =2isshowntogetherwiththelatticedata(blacksquare).
f
low-energy constants (LEC)S andF aswellasthenext-to-leading orderLECL . Thepiondecay
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constant controls thesizeofthenext-to-leading order corrections. Inthispreliminary analysis, we
fixittoanominalvalueF =90MeV.
We fit the value of r¯[0,d ]=(1/d ) d dlr (l ) with the one-loop c PT expression with d =
0
0.01. Namely, both the lattice data and one-loop c PT are integrated in the same region. This
R
valueofd corresponds tothescaleofpionmassofd S /F2≃300MeV,forwhichoneexpectsthat
one-loop c PTconverges reasonably well.
Chiralextrapolationofr [0,0.01]isshowninFigure4. Theone-loopc PTcurveshowsaslight
curvature due to the chiral logarithm. The fit yields S 1/3 = 262.0(1.7) MeV and L = 0.00031(7)
6
withc 2/dof=1.13. Withtheseparametervalues,wedrawthecurvesofr (l )foreachquarkmass
in Figure 3. They explain the rise near l =0 for larger quark masses, but the data beyond al ∼
0.015cannotbeexplained byone-loop c PT.
RenormalizingtotheMSschemeat2GeV,weobtain[S (2GeV)]1/3=260.0(1.7)MeV,where
weusetherenormalizationfactorZ (2GeV)determinedfromtheanalysisofshort-distancecurrent
S
correlator [10].
Probably because the strange quark is too heavy to apply one-loop c PT, a fit with the 2+1-
flavor c PTformulafailedtoreproduce thelatticedata.
5. Discussions
TheChebyshev filtering technique allows precise evaluation of the eigenvalue count in asuf-
ficiently small bin to calculate the eigenvalue spectrum. The method is especially suitable for the
4D effective operator of the domain-wall fermion since the eigenvalue of D(4)†D(4) is limited in
[0,1]. Forthe Wilson fermion, the range is [0,64] (in the freetheory), and one needs much higher
orderpolynomialtoobtainthesameprecision. Thiswouldnearlycompensatethenumericaleffort
toconstruct the(expensive) 4DeffectiveDiracoperatorfromdomain-wallfermion.
Thispreliminaryanalysishasbeendoneusingpartialdataoutofthefulldatasetatthreelattice
spacings and various sea quark masses. We plan to include the data on finer lattices that allow us
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StochasticcalculationofQCDDiracspectrum S.Hashimoto
toextrapolate tothecontinuum limit.
We are grateful to Julius Kuti for private communications on the technique employed in this
work. Theirownworkwasalsopresentedatthisconference. Numericalcalculationwasperformed
on the Blue Gene/Q supercomputer at High Energy Accelerator Research Organization (KEK)
under a support of its Large Scale Simulation Program (No. 14/15-10). The code set Iroiro++
[14], which ishighly optimized for BlueGene/Q, is used. Thiswork issupported in part byJSPS
KAKENHIGrantNumber25800147, 26247043 and15K05065.
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