Multipoint reweighting method and beta functions for the calculation of QCD equation of state

0.16 MB·English

by  Ryo Iwami

Multipoint reweighting method and beta functions
5 for the calculation of QCD equation of state
1
0
2

n
a
J

6 RyoIwami∗1, S.Ejiri2, K.Kanaya3, Y.Nakagawa1,T.Umeda4, D.Yamamoto1
2
(WHOT-QCDCollaboration)

]
t 1GraduateSchoolofScienceandTechnology,NiigataUniversity,Niigata950-2181,Japan
a
l E-mail:[email protected]
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p 2DepartmentofPhysics,NiigataUniversity,Niigata950-2181,Japan
e 3GraduateSchoolofPureandAppliedSciences,UniversityofTsukuba,Tsukuba,Ibaraki
h
[ 305-8571,Japan
4GraduateSchoolofEducation,HiroshimaUniversity,Hiroshima739-8524,Japan
1
v
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We study a reweightingmethodaiming at numericalstudies of QCD at finite density, in which
3
3 the conventional Monte-Carlo method cannot be applied directly. One of the most important
6
problemsinthereweightingmethodistheoverlapproblem. Tosolveit, weproposetoperform
0
. simulations at several simulation points and combine their results in the data analyses. In this
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0 report, we introduce this multipoint reweighting method and test if the method works well by
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measuringhistogramsofphysicalquantities. Usingthismethod,wecalculatethemesonmasses
1
: as continuousfunctionsof the gaugecouplingb andthehoppingparametersk in QCD at zero
v
i density. Wethendeterminelinesofconstantphysicsinthe(b ,k )spaceandevaluatethederiva-
X
tivesofthelatticespacingwithrespecttob andk alongthelinesofconstantphysics(inverseof
r
a thebetafunctions),whichareneededinacalculationoftheequationofstate.
The32ndInternationalSymposiumonLatticeFieldTheory
23-28June,2014
ColumbiaUniversityNewYork,NY
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
MultipointreweightingmethodandbetafunctionsforthecalculationofQCDequationofstate RyoIwami
1. Introduction
In a study of QCD phase diagram at finite temperature (T) and density, the complex quark
determinant causes a serious problem in numerical simulations. The reweighting method is com-
monlyusedtoavoidthisprobleminthelowdensityregion. However,whenweincreasethechemi-
calpotential(m ),thesignproblemandtheoverlapproblembecomessevere. Inthisreport,wefocus
on the overlap problem. The overlap problem is expected to be milder if one changes a couple of
parametersatthesametime. Forexample,WHOT-QCDcollaborationinvestigatedthephasestruc-
ture ofN-flavor QCDin the heavy-quark region, and found that the system at large quark masses
f
is controlled by only two combinations of parameters, b +48(cid:229) Nf k 4 and (cid:229) Nf k Ntcosh(m /T)
f=1 f f=1 f f
on an N3×N lattice, where b = 6/g2, and k and m are the hopping parameter and chemical
s t f f
potential of fth-flavor, respectively [1]. This means that, when one changes the parameters with
keeping these combinations constant, the system does not change, thus the overlap problem does
notarise. Weexpectsuchcombinations ofparametersexistalsointhelight-quark region.
Inthatstudyintheheavy-quarkregion[1],themultipointreweightingmethod[2]forb played
animportantrole: Combiningdataobtainedatdifferentb ,wecouldcalculatetheeffectivepotential
inawiderangeoftheobservablevalues,thatwasmandatoryinareliableevaluationofthetransition
point. In the present paper, we extend the multipoint reweighting method in the multi-parameter
spaceofb andk toovercometheoverlapprobleminthelight-quark region.
f
Inthenextsection,themultipointreweightingmethodisintroduced,and,inSec.3,weexamine
the overlap problem performing numerical simulations in two-flavor QCD. We then calculate the
meson masses, thelines ofconstant physics, andthederivatives ofthelattice spacing withrespect
tob andk alongthelinesofconstant physicsinSec.4. Thesection5isfortheconclusions.
2. Multipoint reweighting method
We extend the multipoint reweighting method to the multi-parameter space such that both
gaugeandquarkparametersarechangedsimultaneously. Letusfirstdefineahistogramforasetof
physical quantities X =(X ,X ,···)as
1 2
w(X;b ,~k ,~m ) = DU (cid:213) d (X −Xˆ)e−Sg (cid:213)Nf detM(k ,m ). (2.1)
i i f f
Z i f=1
where S is the gauge action, M is the quark matrix and Xˆ =(Xˆ ,Xˆ ,···) is the operators for X.
g 1 2
WedenoteS≡S −(cid:229) lndetM. Thecouplingparametersofthetheoryareb ,~k =(k ,k ,···,k ),
g f 1 2 Nf
and~m =(m ,m ,···,m ). Then, thepartition function isgivenbyZ(b ,~k ,~m )= w(X;b ,~k ,~m )dX
1 2 Nf
withdX =(cid:213) dX,andtheprobability distribution functionofX isgivenbyZ−1w(X;b ,~k ,~m ). The
i i R
expectation valueofanoperator O[Xˆ]whichiswrittenintermsofXˆ iscalculated by
1
hO[Xˆ]i(b ,~k ,~m )= Z(b ,~k ,~m ) O[~X]w(~X;b ,~k ,~m )dX. (2.2)
Z
For simplicity, we denote the set of coupling parameters (b ,~k ,~m ) as b in the following of this
section.
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MultipointreweightingmethodandbetafunctionsforthecalculationofQCDequationofstate RyoIwami
We perform a simulation at b and calculate the histogram at b . We fix three quantities X,
0
S(b )≡S and S(b )≡S to construct the histogram, where S is the value of the action with the
0 0
coupling parameters b ontheconfiguration generated atb . FromEq.(2.1),wefind
0
w(X,S,S ;b )=e−(S−S0)w(X,S,S ;b ). (2.3)
0 0 0
Thehistogram ofX isgivenbyw(X;b )= w(X,S,S ;b )dSdS .
0 0
When X has large correlation with the difference of the actions, S−S , the peak position of
R 0
the distribution may change appreciably, causing the overlap problem. (SeeSec.3.) Toovercome
the overlap problem and to obtain w and V that are reliable in a wide range of X, we extend
eff
thereweighing formulas tocombine dataobtained atdifferent simulation points [1,2]forthecase
including thefermionaction.
We combine a set of N simulations performed at b with the number of configurations N
sp i i
wherei=1,···,N . UsingEq.(2.3),theprobability distribution function atb isrelatedtothatat
sp i
b as
Z−1(b )w(X,S,~S;b )=Z−1(b )e−(Si−S)w(X,S,~S;b ). (2.4)
i i i
Summinguptheseprobability distribution functions withtheweightN,
i
Nsp Nsp
(cid:229) N Z−1(b )w(X,S,~S;b )=eS(cid:229) N Z−1(b )e−Siw(X,S,~S;b ), (2.5)
i i i i i
i=1 i=1
weobtain
Nsp
w(X,S,~S;b )=G(S,~S;b ,~b ) (cid:229) N Z−1(b )w(X,S,~S;b ) (2.6)
i i i
i=1
where~b =(b ,···,b )and
1 Nsp
e−S
G(S,~S;b ,~b )= . (2.7)
(cid:229) Ni=sp1Nie−SiZ−1(b i)
Notethat theleft-hand sideofEq.(2.5) givesanaivehistogram using allthe configurations disre-
garding the difference in the simulation parameter (b ,~k ). Thehistogram w(X,S,~S;b ) at (b ,~k ) is
givenbymultiplying G(S,~S;b ,~b )tothenaivehistogram.
Thepartition functionisgivenby
Nsp Nsp
Z(b )= (cid:229) N G(S,~S;b ,~b )Z−1(b )w(X,S,~S;b )dXdSd~S= (cid:229) N G(S,~S;b ,~b ) . (2.8)
i i i i
i=1 Z i=1 (bi)
D E
The right-hand side is just the naive sum of G(S,~S;b ,~b ) obtained on all the configurations. The
partition function atb canbedetermined, uptoanoverallfactor, bytheconsistency relations,
i
Z(b )= (cid:229)Nsp N G(S,~S;b ,~b ) = (cid:229)Nsp N e−Si (2.9)
i k=1 kD i E(bk) k=1 k*(cid:229) Nj=sp1Nje−SjZ−1(b j)+(bk)
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MultipointreweightingmethodandbetafunctionsforthecalculationofQCDequationofstate RyoIwami
fori=1,···,N . Denoting f =−lnZ(b ),theseequations canberewrittenby
sp i i
1=(cid:229) Nsp N (cid:229) Nsp N exp[S −S − f + f ] −1 . Starting from appropriate initial values of
k=1 k j=1 j i j i j
(cid:28)(cid:16) (cid:17) (cid:29)(bk)
f,wesolve theseequations numerically byaniterative method. Notethat, inthiscalculation, one
i
of f mustbefixedtoremovetheambiguitycorresponding totheundetermined overallfactor.
i
Theexpectation valueofanoperatorXˆ atb canbeevaluatedas
hXˆi(b ) = Z(1b )Z Xw(X,S,~S;b )dSd~S = Z(1b )i(cid:229)N=sp1Ni XˆG(S,~S;b ,~b ) (bi), (2.10)
D E
andthehistogram ofX isobtained by
Nsp
w(X;b ) = (cid:229) N d (X−Xˆ)G(S,~S;b ,~b ) , (2.11)
i
i=1 (bi)
D E
Note that, again, (cid:229) Nsp N XˆG in the right-hand side is just the naive sum of XG over all the
i=1 i (bi)
configurations disregarding thedifferenceinthesimulationpointb .
(cid:10) (cid:11) i
3. Overlapproblem andhistograms
To study if the multipoint reweighting method is useful in avoiding the overlap problem, we
perform simulations ofQCDwithdegenerate 2flavors ofclover-improved Wilson quarks coupled
withRG-improvedIwasakiglue,atm =0. Theimprovementparametersoftheactionarethesame
asthoseadopted inRef.[4]. Thesimulations arecarried outonan84 latticeat9simulation points
(3b ’s×3k ’s)fortheteststudyinthissection, andona164 latticeat30points(6b ’s×5k ’s)in
Sec.4. Thenumberofconfigurations forthemeasurementis200ateachsimulationpoint.
The action S required in the reweighting method is composed by Wilson loops and lndetM.
The calculation of the fermion part requires large computational cost. In this study, we evaluate
detM by measuring the first and second derivatives of lndetM at several k ’s, where i=1,2,···,
i
on each configuration, and interpolate the determinant between k and k assuming a quadratic
i i+1
function, lndetM(k )=lndetM(k )+C (k −k )+C (k −k )2+C (k −k )3+C (k −k )4. The
i 1 i 3 i 3 i 4 i
coefficients C are determined such that the first and second derivatives are consistent with the
a
measured valuesatk andk . Theclovertermwiththecoefficient c isalsoevaluated interms
i i+1 SW
of its derivatives. Here, because c depends on b in our choice, it affects the b -dependence of
SW
observables. However, we find that the effects from c are very small in the range of b rele-
SW
vant in our study. We thus approximate the b -dependence of the action by a linear function, i.e.
lndetM(b ,~k )=lndetM(b ,~k )+(b −b )(dc /db )(¶ lndetM/¶ c )(b ,~k ) [3]. Note that the
0 0 SW SW 0
overallconstant oflndetM isnotneeded.
In the left panel of Fig. 1, we show the results for the improved plaquette P = c W1×1+
0
2c W1×2 of the Iwasaki action at b =1.825, where c =−0.331, c =1−8c , and Wi×j is the
1 1 0 1
(i× j)Wilsonloop. Blackdotsrepresenttheexpectation valuesofPatthethreesimulation points
withoutreweighting. Blue,greenandpurplecurvesaretheresultsofthenaivereweightingmethod
usingthedataatk =0.1400,0.1425,and0.1440,respectively. Eachresultofthenaivereweighting
method is reliable around the corresponding simulation point, but fails reproducing far away sim-
ulation results. The reason can be easily understood by consulting the histogram of P: Red curve
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MultipointreweightingmethodandbetafunctionsforthecalculationofQCDequationofstate RyoIwami
0.5 0.5
k =0.1400 k =0.1400
1.7 k =0.1400 k =0.1412 k =0.1412
kk ==00..11442450 0.4 kk ==00..11442346 0.4 kk ==00..11442346
simulation point k =0.1440 k =0.1440
1.69 multi point
0.3 0.3
P
1.68 0.2 0.2
0.1 0.1
1.67
0 0
1.64 1.66 1.68 1.7 1.72 1.64 1.66 1.68 1.7 1.72
1.66 0.14 0.1k 42 0.144 P P
Figure1: Left: TheexpectationvalueofP≡c W1×1+2c W1×2 asafunctionofk atb =1.825. Middle:
0 1
The histogram of P at various k ’s obtained by the naive reweighting method using the configurations at
k =0.140.Right:ThehistogramofPbythemultipointreweightingmethod.
(b ,k )=(1.8325,0.14000) (b ,k )=(1.8250,0.14000)
(b ,k )=(1.8400,0.14000) (b ,k )=(1.8250,0.14160)
(b ,k )=(1.8475,0.14000) (b ,k )=(1.8250,0.14240)
0.02 0.02
0.016 0.016
0.012 0.012
0.008 0.008
0.004 0.004
0 0
4 9.6 4 9.6
3 3
9.8 9.8
2 2
[dS / dk ] 1 10 dS / db [dS / dk ] 1 10 dS / db
SUB 0 SUB 0
-1 -1
Figure 2: The b -dependence (left) and k -dependence (right) of the histogram for N−1(¶ S/¶b ) and
site
N−1[¶ S/¶k ] ≡N−1[¶ S/¶k −(288Nk 4/c )(¶ S/¶b )],whereN =84.
site SUB site f 0 site
in the middle panel ofFig. 1 isthe original histogram at(b ,k )=(1.825,0.140), and green, blue,
magentaandlightbluecurvesarethehistogramsatk =0.1412,0.1424,0.1436,and0.144,respec-
tively, estimated by the naive reweighting method Eq. (2.3) using the data at k =0.140. Because
the histograms atk other than the simulation point is calculated asthe product of the reweighting
factor and the original histogram, the histograms are not reliable out of the range of the original
distribution. Infact,thevalueofPdistributes between1.64and1.69. Evenwhenk ischanged by
the reweighting method, the upper and lower bounds of the distribution does not change in Fig. 1
(middle). Since the expectation value is approximately the peak position of the histogram, the
expectation valuealsocannot gooutoftherangeofthedistribution, asshowninFig.1(left).
To enlarge the range of the distribution, we combine the simulation data obtained at k =
0.1400, 0.1425 and 0.1440 using the multipoint reweighting method explained in the previous
section. TheredcurveintheleftpanelofFig.1istheresultofthemultipointreweightingmethod.
Wefindthattheredcurvesmoothlyconnectsallthedirectsimulationresultswithsmallerrorbars.
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MultipointreweightingmethodandbetafunctionsforthecalculationofQCDequationofstate RyoIwami
b =1.806 mPS / mV = 0.70
0.8 b =1.8125 0.1435 mPS / mV = 0.72
b =1.819 mPS / mV = 0.74
b =1.825 mPS / mV = 0.76
b =1.831
b =1.837 0.143
0.75
V
m
/ PS k0.1425
m
0.7
0.142
0.65 0.1415
6.95 7 7.05 7.1 7.15 1.815 1.82 1.825 1.83 1.835
1 / k b
Figure3:Left:Thepseudoscalarandvectormesonmassratiom /m .Right:Thelinesofconstantphysics
PS V
foreachm /m inthe(b ,k )plane.
PS V
In the right panel of Fig. 1, histograms from the multipoint reweighting method are plotted for
k =0.1412, 0.1424,0.1436, and0.144.
Themethodisapplicable tootherobservables. Theexpectation valuesof¶ S/¶b and
[¶ S/¶k ] ≡¶ S/¶k −(288N k 4/c )(¶ S/¶b )
SUB f 0
are needed in the calculation of the equation of state in the integral method. In the right and
left panels of Fig. 2, we show the b - and k -dependence of the 2-dimensional histogram of these
quantities using the multipoint reweighting method. The histogram moves as b and k are varied.
We can compute the expectation values of ¶ S/¶b and [¶ S/¶k ] as continuous functions of b
SUB
andk withouttheoverlapproblem. Onceweobtainphysical quantities ascontinuous functions of
couplingparameters,wecancalculatethelinesofconstantphysicsinthecouplingparameterspace
aswellasthebetafunctions.
4. Lines ofconstant physics andbeta functions
In this study, we define the lines of constant physics by fixing the dimension-less ratio of
pseudoscalar and vector meson masses m /m in the (b ,k ) space. On the lattice, we measure
PS V
dimension-less observables m a and m a, where a is the lattice spacing. Because these lattice
PS V
observables varyaswechange b ork ,aisvariedwhenwemovealongalineofconstant physics.
Thebeta functions ¶b /¶ a and¶k /¶ aare defined through thevariation aalong alineofconstant
physics. Thebetafunctions areneededinthecalculation oftheequation ofstate.
Themultipoint reweighting methodisuseful forthecalculation ofthebeta functions because
wecancalculate observables ascontinuous functions ofb andk . Combining thedataat30simu-
lationpoints(6b ’s×5k ’s)onthe164 latticebythemultipointreweightingmethod,weobtainthe
massratiom /m plotted intheleftpanel ofFig.3. Fromthese data, determine thelinesofcon-
PS V
stantphysicsform /m =0.70,0.72,0.74and0.76,asshowninFig.3(right). Wethencalculate
PS V
thederivatives,(m a)¶b /¶ (m a)and(m a)¶k /¶ (m a)alongeachlineofconstantphysics. The
V V V V
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MultipointreweightingmethodandbetafunctionsforthecalculationofQCDequationofstate RyoIwami
0 0.04
-0.5 0.02
a) a)
mv mV
/d -1 /d 0
bma(dv kma(dV
m / m = 0.70
m / m = 0.70 PS V
-1.5 mPS / mV = 0.72 -0.02 mPS / mV = 0.72
PS V m / m = 0.74
m / m = 0.74 PS V
PS V m / m = 0.76
m / m = 0.76 PS V
PS V
-2 -0.04
1.816 1.82 1.824 1.828 1.832 1.816 1.82 1.824 1.828 1.832
b b
Figure4: Thebetafunctions:(m a)¶b /¶ (m a)(left)and(m a)¶k /¶ (m a)(right).
V V V V
resultsareshowninFig.4. Combiningthesebetafunctionswiththemeasurementofh¶ S/¶k iand
h¶ S/¶b ionfinite-temperature lattice,wecancalculate theequation ofstate.
5. Conclusions andoutlook
We discussed the multipoint reweighting method in a multi-dimensional parameter space to
avoid the overlap problem. Using the method, we can reliably calculate histograms of physical
quantities as well as the expectation values of the physical quantities, as continuous functions
of coupling parameters. These enable us to compute the lines of constant physics and the beta
functions, whichareneededinacalculation oftheequationofstate.
Ourfinalobjective isastudyoffinitedensity QCD.Usingthemultipoint reweighing method,
we may absorb the main effect of the chemical potential by a change of b and k . If so, we may
investigate finitedensity QCDavoiding thesignproblem.
Acknowledgments Thiswork isinpart supported byGrants-in-Aid ofthe Japanese Ministry of
Education, Culture,Sports,ScienceandTechnology (Nos.26400244, 26400251).
References
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