Lattice and string worldsheet in AdS/CFT: a numerical study

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by  Valentina Forini

HU-EP-16/04
QMUL-PH-16-01
Lattice and string worldsheet in AdS/CFT:
a numerical study
Valentina Forini
∗
6 Humboldt-UniversitätzuBerlin,ZumGroßenWindkanal6,12489Berlin,Germany
1 E-mail: [email protected]
0
2 Lorenzo Bianchi
n Humboldt-UniversitätzuBerlin,ZumGroßenWindkanal6,12489Berlin,Germany
a II.InstitutfürTheoretischePhysik,UniversitätHamburg,LuruperChaussee149,22761
J Hamburg,Germany

8 E-mail: [email protected]
1
Marco S. Bianchi

] CenterforResearchinStringTheory-SchoolofPhysicsandAstronomy
t
a QueenMaryUniversityofLondon,MileEndRoad,LondonE14NS,UK
l E-mail: [email protected]
-
p
Bjoern Leder
e
h Institutsrechenzentrum,InstitutfürPhysik
[ Humboldt-UniversitätzuBerlin,Newtonstr. 15,12489Berlin,German

E-mail: [email protected]
1
v
Edoardo Vescovi
0
7 Humboldt-UniversitätzuBerlin,ZumGroßenWindkanal6,12489Berlin,Germany
6 E-mail: [email protected]
4
0
We consider a possible discretization for the gauge-fixed Green-Schwarz (two-dimensional)
.
1 sigma-model action for the Type IIB superstring and use it for measuring the cusp anomalous
0 dimensionofplanarN =4SYMasderivedfromstringtheory. Weperformlatticesimulations
6
employing a Rational Hybrid Monte Carlo (RHMC) algorithm and a Wilson-like fermion dis-
1
: cretization. Inthispreliminarystudy,wecompareourresultswiththeexpectedbehaviorforvery
v
large values of g= √λ, which is the perturbative regime of the sigma-model, and find a qual-
i 4π
X itative agreement at finite lattice spacing. For smaller g the continuum limit is obstructed by a
r divergence. Wealsodetectaphaseinthefermiondeterminant,whoseoriginweexplain,which
a
forsmallgleadstoasignproblemnottreatableviastandardreweigthing. Resultspresentedhere
arediscussedthoroughlyin[1].
The33rdInternationalSymposiumonLatticeFieldTheory
14-18July2015
KobeInternationalConferenceCenter,Kobe,Japan*
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
LatticeandstringworldsheetinAdS/CFT:anumericalstudy ValentinaForini
1. Discussion
Animpressiveamountofevidenceobtainedinthelastdecadeindicatesthemaximallysuper-
symmetricandsuperconformalN =4planarsuperYang-Mills(SYM)theoryasthefirstexample
of a non-trivially interacting, four-dimensional gauge theory that might be exactly solvable. The
“evidence”referstotheseveralresults,obtainedrelyingontheassumptionofanall-orderquantum
integrabilityforthismodel[2],thathavebeenconfirmedbydirectperturbativecomputationsboth
ingaugetheoryandinwhatAdS/CFTassignsasstringdualtoN =4SYM-theTypeIIB,Green-
SchwarzstringpropagatinginthemaximallysupersymmetricbackgroundAdS S5 supportedby
5
×
a self-dual Ramond-Ramond (RR) five-form flux. However, without the assumption of quantum
integrability, very little is known about the theory at finite coupling. Here a lattice approach may
bepowerful. ArichandinterestingprogramofputtingN =4SYMonthelatticeisbeingcarried
outforsomeyearsbyCatteralletal.[3]1.
Followingtheearlierproposalof[5], wetakeadifferentlattice-basedroutetoinvestigatethe
finite coupling region of the N = 4 SYM theory, discretizing the dual two-dimensional string
worldsheet. We focus on a particularly important observable of the theory – the cusp anomalous
dimension–whichwedescribeinSection2. FromthedualpointofviewprovidedbytheAdS/CFT
correspondence [6], this quantity is measured by the path integral of an open string bounded by a
cuspedWilsonloopattheAdSboundary,whichiswherethefour-dimensionalgauge-theorylives
intheholographicpicture. Thisstringworldsheetisahighlynon-trivial2dnon-linearsigma-model
withrichnon-perturbativedynamics.
We propose a discretization of such a model in Section 3. It is important to emphasize that,
sincethegaugesymmetriesofthemodel(bosonicdiffeomorphismsandκ-symmetry)areallfixed,
thisisnotadefinitionoftheGreen-SchwarzworldsheetstringmodelàlaWilsonlattice-QCD,but
ratheraninvestigationofpossibleroutesviawhichlatticesimulationscouldbeanefficienttoolin
numericalholography.
In Section 4 we describe our lattice simulations, which use the Rational Hybrid Monte Carlo
(RHMC) algorithm. Our line of constant physics demands physical masses to be kept constant
while approaching the continuum limit, which in the case of finite mass renormalization requires
no tuning of the “bare” mass parameter of the theory (the light-cone momentum P+). For one of
thebosonicfieldsenteringtheLagrangianwedeterminethecorrelatorandphysicalmass,confirm-
ing the expected finite renormalization and thus no need of tuning. A good agreement with the
perturbativesigma-modelexpectationisreachedforboththecorrelatorsandtheobservableunder
investigation at very large values of the coupling g=√λ/(4π) (λ is the ’t Hooft coupling of the
AdS/CFT dual gauge theory) 2. For smaller g we observe a divergence, possibly signaling the
presence of infinite mass renormalization for the field content of the theory which so far we have
notanalyzedandwhosestudyweleaveforthefuture.
Oneimportantresultofouranalysisisthedetectionofaphaseinthefermionicdeterminant,re-
sultingfromintegratingoutthefermions. Thisphaseisintroducedbythelinearizationoffermionic
interactionsusedin[5]. Forvaluesofthecouplingapproachingthenon-perturbativeregime(cor-
responding to weakly-coupled N =4 SYM) the phase undergoes strong fluctuations, signaling a
severe sign problem. It would be desirable to find alternative ways to linearize quartic fermionic
interactions,withresultingYukawatermsleadingtoareal,positivedefinitefermionicdeterminant.
Attemptsinthisdirectionareongoingandwehopetoreportontheminthenearfuture.
2. Theobservable: cuspanomalyofplanarN =4SYMfromstringtheory
The cusp anomaly of N = 4 SYM, in this framework often simply referred to as “scaling
1WereferthereadertothePOScontribution[4]forareportoninterestingrecentresults.
2In the AdS/CFT context, where the ’t Hooft coupling λ g2 is used as relevant parameter, the large g region
∼
is naturally referred to as “strong coupling” regime. The string worldsheet sigma-model of interest here, for which
perturbationtheoryisa1/gexpansion,ishoweverweakly-coupledatlargeg.
2
LatticeandstringworldsheetinAdS/CFT:anumericalstudy ValentinaForini
function”3,isafunctionofthecouplinganditgovernstherenormalizationofaWilsonloopwith
a light-like cusp in N =4 super Yang-Mills, as well as the leading behavior, in the large spin S
regime, of the anomalous dimension of twist-two operators. According to AdS/CFT, any Wilson
loop expectation value should be represented by the path integral of an open string ending at the
AdSboundary
(cid:90)
W[Ccusp] Zcusp= [DδX][DδΨ]e−Scusp[Xcl+δX,δΨ]=e−Γeff e−18f(g)V2 . (2.2)
(cid:104) (cid:105)≡ ≡
Above,X =X (t,s)-witht,sthetemporalandspatialcoordinatespanningthestringworldsheet
cl cl
-istheclassicalsolutionofthestringequationsofmotiondescribingtheworldsurfaceofanopen
stringendingonanullcusp[8]. S [X+δX,δΨ]istheactionforfieldfluctuationsoverit–the
cusp
fieldsbeingbothbosonicandfermionicstringcoordinatesX(t,s), Ψ(t,s)–andisreportedbelow
inequation(2.4)intermsoftheeffectivebosonicandfermionicdegreesoffreedomremainingaf-
tergauge-fixing. SincethefluctuationLagrangianhasconstantcoefficients,theworldsheetvolume
V =(cid:82) dtdssimplyfactorizesout4infrontofthefunctionofthecoupling f(g),asinthelastequiv-
2
alencein(2.2). Thescalingfunctioncanbeevaluatedperturbativelyingaugetheory(g 1),and
(cid:28)
in sigma-model loop expansion (g 1) as in (2.5) below. Assuming all-order integrability of the
(cid:29)
spectralproblemfortherelevantoperatorsandtakingathermodynamicallimitofthecorrespond-
ing asymptotic Bethe Ansatz, an integral equation [9] can be derived which gives f(g) exactly at
eachvalueofthecoupling.
Ratherthanpartitionfunctions,inalatticeapproachitisnaturaltostudyvacuumexpectation
values. Insimulatingthevacuumexpectationvalueofthe“cusp”action
(cid:82)[DδX][DδΨ]Scuspe−Scusp dlnZcusp V2
S = = g g f (g), (2.3)
(cid:104) cusp(cid:105) (cid:82)[DδX][DδΨ]e Scusp − dg ≡ 8 (cid:48)
−
wearethereforesupposedtoobtaininformationonthederivativeofthescalingfunction5.
In the continuum, the AdS S5 superstring action S describing quantum fluctuations
5 cusp
×
aroundthenull-cuspbackgroundcanbewrittenafterWick-rotationas[8]
S =g(cid:90) dtds(cid:110) ∂ x+1x 2+ 1 ∂ x 1x 2+(cid:16)∂ zM+1zM+ i z η (cid:0)ρMN(cid:1)i ηj(cid:17)2
cusp | t 2 | z4| s −2 | t 2 z2 N i j
+1 (cid:0)∂ zM 1zM(cid:1)2+i(cid:0)θi∂ θ +ηi∂ η +θ∂ θi+η∂ ηi(cid:1) 1 (cid:0)ηiη(cid:1)2 (2.4)
z4 s −2 t i t i i t i t −z2 i
(cid:104) (cid:16) (cid:17)(cid:105)(cid:111)
+2i z13zMηi(cid:0)ρM(cid:1)ij(cid:0)∂sθj−21θj−ziηj(cid:0)∂sx−12x(cid:1)(cid:1)+z13zMηi(ρM†)ij ∂sθj−12θj+ziηj(cid:0)∂sx−12x(cid:1)∗
Above, x,x are the two bosonic AdS (coordinate) fields transverse to the AdS subspace of the
∗ 5 3
classicalsolution,andzM(M=1, ,6)arethebosoniccoordinatesoftheAdS S5backgroundin
5
···(cid:112) ×
Poincaréparametrization,withz= z zM,remainingafterfixinga“AdSlight-conegauge”[10].
M
3The“scalingfunction” f(g)isinfactthecoefficientoflogSinthelargespinSanomalousdimension∆ofleading
twistoperators∆= f(g)logS+O(logS/S). ItequalstwicethecuspanomalousdimensionΓcusp oflight-likeWilson
loops[7]
W[Ccusp] e−ΓcuspγlnΛΛUIRV , (2.1)
(cid:104) (cid:105)∼
whereγisthelarge,realparameterrelatedtothegeometricangleφofthecuspedWilsonloopbyiγ=φ.Theexpectation
valueaboveisinfactextractedinthelargeimaginaryφ limit.
4Asmentionedabove, f(g)equalstwicethecoefficientofthelogarithmicdivergencein(2.1),forwhichthestringy
counterpartshouldbetheinfinitetwo-dimensionalworldsheetvolume.ThefurthernormalizationofV2witha1/4factor
followstheconventionof[8].
5Hereouranalysisisdifferentfromtheonein[5]. Inparticular, S f(g) onlywhen f(g)islinearing,which
(cid:104) (cid:105)∼ V2/2
happensasfrom(2.5)forlargeg.
3
LatticeandstringworldsheetinAdS/CFT:anumericalstudy ValentinaForini
Thefieldsθ,η,i=1,2,3,4are4+4complexanticommutingvariablesforwhichθi=(θ)†,ηi=
i i i
(η)†. TheytransforminthefundamentalrepresentationoftheSU(4)R-symmetryanddonotcarry
i
(Lorentz)spinorindices. ThematricesρM aretheoff-diagonalblocksofSO(6)DiracmatricesγM
ij
in the chiral representation and (ρMN) j =(ρ[Mρ†N]) j are the SO(6) generators. The action (2.4)
i i
ismanifestlymissingamassiveparameter6,whichwerestoreinthefollowingdefiningitasm. We
emphasizethat,in(2.4),localbosonic(diffeomorphism)andfermionic(κ-)symmetriesoriginally
presenthavebeenfixed. Withthisactiononecandirectlyproceedtotheperturbativeevaluationof
theeffectiveactionin(2.2),uptotwoloopsinsigma-modelperturbationtheory[8],obtainingfor
thecuspanomaly(K istheCatalanconstant)
(cid:16) 3log2 K (cid:17)
f(g)=4g 1 +O(g 3) . (2.5)
− 4πg −16π2g2 −
Furthermore,withthesameactionitispossibletostudyperturbativelythe(non-relativistic)disper-
sionrelationforthefieldexcitationsovertheclassicalstringsurface. Forexample,thecorrections
to the masses of the bosonic fields x,x in (2.4) (defined as the values of energy at vanishing mo-
∗
mentum)read[11]
m2(cid:16) 1 (cid:17)
m2(g)= 1 +O(g 2) , (2.6)
x 2 −8g −
where, as mentioned above, we restored the dimensionful parameter m. In what follows, we will
computethelatticecorrelatorsofthefieldsx,x sotostudywhetherourdiscretizationchangesthe
∗
renormalizationpatternabove.
3. Linearizationanddiscretization
Whilethebosonicpartof(2.4)canbeeasilydiscretizedandsimulated, Graßmann-oddfields
are formallyintegrated out, lettingtheir determinant tobecome part –via exponentiation interms
of pseudo-fermions, see (3.6) below – of the Boltzmann weight of each configuration in the sta-
tistical ensemble. In the case of higher-order fermionic interactions – as in (2.4), where they are
at most quartic – this is possible via the introduction of auxiliary fields realizing a linearization.
Following[5],oneintroduces7realauxiliaryfields,onescalarφ andaSO(6)vectorfieldφ ,with
M
aHubbard-Stratonovichtransformation
exp(cid:110) g(cid:90) dtds(cid:104) 1 (cid:0)ηiη(cid:1)2+(cid:16) i z ηρMNi ηj(cid:17)2(cid:105) (3.1)
− −z2 i z2 N i j }
(cid:90) (cid:110) (cid:90) (cid:111)
DφDφM exp g dtds[1φ2+√2φη2+1(φ )2 i√2φMz (cid:0)iηρMNi ηj(cid:1)] .
∼ − 2 z 2 M − z2 N i j
Above, in the second line we have written the Lagrangian for φM so to emphasize that it has an
imaginarypart,duetothefactthatthebilinearforminroundbracketsishermitian
(cid:16)iηρMNi ηj(cid:17)†= i(ηj)†(ρMNi ) (η)†= iη ρMN jηi=iη ρMNj ηi, (3.2)
i j j ∗ i j i j i
− −
asfollowsfromthepropertiesoftheSO(6)generators(seeforexampleAppendixAof[8]). Since
theauxiliaryvectorfieldφM hasrealsupport,theYukawa-termforitsetsaprioriaphaseproblem,
the only question being whether the latter is treatable via standard reweighing. Below we find
evidencethatthisisnotthecase.
6Asstandardintheliterature-thelight-conemomentumcanbeconsistentlysettotheunitaryvalue,p+=1.Inthe
perspectiveadoptedhere,however,itiscrucialtokeeptrackofdimensionfulparametersastheyareinprinciplesubject
torenormalization.
4
LatticeandstringworldsheetinAdS/CFT:anumericalstudy ValentinaForini
Afterthetransformation(3.1),thecorrespondingLagrangianreads
L = ∂ x+mx 2+1 ∂ x mx 2+(∂ zM+1zM)2+1(∂ zM mzM)2+1φ2+1(φ )2+ψTO ψ(3.3)
| t 2 | z4| s −2 | t 2 z4 s −2 2 2 M F
withψ (θi,θ,ηi,η)and
i i
≡
 0 i∂ iρM(cid:0)∂ +m(cid:1)zM 0 
t − s 2 z3
 i∂ 0 0 iρ† (cid:0)∂ +m(cid:1)zM 
O =  t − M s 2 z3  ,(3.4)
F izMρM(cid:0)∂ m(cid:1) 0 2zMρM(cid:0)∂ x mx(cid:1) i∂ AT 
 z3 s− 2 z4 s − 2 t− 
0 izMρ† (cid:0)∂ m(cid:1) i∂ +A 2zMρ† (cid:0)∂ x mx (cid:1)
z3 M s− 2 t − z4 M s ∗− 2∗
1 1 z
A = φ ρMNz φ +i NρMN∂ zM . (3.5)
√2z2 M N−√2z z2 t
ThequadraticfermioniccontributionresultingfromlinearizationgivesthenformallyaPfaffian
PfO , which - in order to enter the Boltzmann weight and thus be interpreted as a probability -
F
shouldbepositivedefinite. Forthisreason,weproceedasin[5]
(cid:90) DΨe−(cid:82) dtdsΨTOFΨ=PfOF ≡(detOFO†F)14 =(cid:90) DξDξ¯e−(cid:82)dtdsξ¯(OFO†F)−41ξ , (3.6)
wherethesecondequivalenceobviouslyignorespotentialphasesoranomaliesandwearenotbeing
explicitincoupling-dependentJacobians(seebelow).
The values of the discretised (scalar) fields are assigned to each lattice site, with periodic
boundary conditions for all the fields except for antiperiodic temporal boundary conditions in the
caseoffermions. Thediscreteapproximationofcontinuumderivativesarefinitedifferenceopera-
torsdefinedonthelattice. AWilson-likelatticeoperatormustbeintroduced[1],suchthatfermion
doublersaresuppressedandtheone-loopconstant 3ln2/π in(2.5)isrecoveredinlatticepertur-
−
bationtheory. WhilesuchWilsontermexplicitlybreakstheSO(6)symmetryofthemodel,forthe
SO(6) singlet quantities that we study – f(g) and x,x correlators – this might affect the way one
∗
approachesthecontinuumlimit,anissuewhichweaddressinthenextSection.
The Monte Carlo evolution of the action (3.3) is generated by the standard Rational Hybrid
Monte Carlo (RHMC) algorithm, as in [5]. A detailed analysis of the precision of the fourth-root
approximationandthemagnitudeofthefermionmatrixeigenvalues(3.4),erroranalysisandauto-
correlationtimesoftheMonteCarlohistoriesisin[1].
4. Simulation,continuumlimitandthephase
As discussed above, in the continuum model there are two parameters, the dimensionless
couplingg= √λ andthemassscalem. Intakingthecontinuumlimit, thedimensionlessphysical
4π
quantitiesthatitisnaturaltokeepconstantarethephysicalmassesofthefieldexcitationsrescaled
by L, the spatial lattice extent. This is our line of constant physics. For the example in (2.6), this
means
L2m2=const, whichleadsto L2m2 (NM)2=const, (4.1)
x
≡
where we defined the dimensionless M =ma with the lattice spacing a. The second equation in
(4.1)reliesfirstontheassumptionthatgisnotrenormalized,whichissuggestedlatticeperturbation
theory [1]. Second, one should investigate whether relation (2.6), and the analogue ones for the
other fields of the model, are still true in the discretized model - i.e. the physical masses undergo
only a finite renormalization. In this case, at each fixed g fixing L2 m2 constant would be enough
tokeeptherescaledphysicalmassesconstant,namelynotuningofthe“bare”parametermwould
benecessary. Inthiscontribution,westartbyconsideringtheexampleofbosonicx,x correlators,
∗
5
LatticeandstringworldsheetinAdS/CFT:anumericalstudy ValentinaForini
1.4 L/a=8,g=10 1.2
L/a=10,g=10
1.2 L/a=8,g=30 1
L/a=10,g=30
1 L/a=12,g=30 /S0
2m 2N) 0.8
eff2m)/x 0.8 >7.5− 0.6
( 0.6 S
< 0.4
(
0.4
g=100.00,Lm=4
0.2 0.2 g=30.00,Lm=4
g=10.00,Lm=4
00 0.2 0.4 0.6 0.8 1 1.2 1.4 00 0.02 0.04 0.06 0.08 0.1 0.12
tmx 1/N
Figure 1: Left panel: Effective mass plot meff = 1ln Cx(t) , as calculated from the correlator
x a Cx(t+a)
Cx(t)=∑s1,s2(cid:104)x(t,s1)x∗(0,s2)(cid:105)ofbosonicfieldsx,x∗ inpresenceofWilsonterms.
Rightpanel: Plotoftheratio (cid:104)SLAT(cid:105)S−cN2/2,whereS0=M2N2g/2andc/2=7.5(1).
0
where indeed we find no (1/a) divergence for the ratio m2/m2 – see the left panel in Figure 1. In
x
thelargegregionthatweinvestigatetheratioconsideredapproachestheexpectedcontinuumvalue
1/2. Having this as hint, and because with the proposed discretization we recover in perturbation
theorytheone-loopcuspanomaly[1],weassumethatinthediscretizedmodelnofurtherscalebut
thelatticespacingaispresent. AnyobservableF isthereforeafunctionF =F (g,N,M)of
LAT LAT LAT
theinput(dimensionless)parametersg= √λ,N= L andM=am. InTable1welisttheparameters
4π a
ofthesimulationspresentedinthispaper. AtfixedcouplinggandfixedmL MN (largeenough
so to keep finite volume effects e mL small), F is evaluated for differe≡nt values of N. The
− LAT
∼
continuumlimit–whichwedonotattempthere–isthenobtainedextrapolatingtoinfiniteN.
Inmeasuringtheaction(2.3)onthelattice,wearesupposedtorecoverthefollowinggeneral
behavior
S c 1
(cid:104) NLA2T(cid:105) = 2+8M2gf(cid:48)(g), (4.2)
wherewehavereinsertedtheparameterm,usedthatV =a2N2 andaddedaconstantcontribution
2
in g which takes into account possible coupling-dependent Jacobians relating the (derivative of
the) partition function on the lattice to the one in the continuum. Measurements for the ratio
(cid:104)SLMAT2(cid:105)N−2cgN/22/2 = f(cid:48)4(g) are, at g=100, in good agreement with 2c =7.5(1) – consistently with the
expectation [1] – and with the leading order prediction in (2.5) for which f (g)=4. For lower
(cid:48)
values of g – red and blue curves in Figure 1, right panel – we observe a deviation that obstructs
thecontinuumlimitandiscompatiblewiththepresenceofadivergence. Thiscouldbeexplained
with the presence of infinite mass renormalization in those fields correlators which we have not
consideredsofar,andwhoseinvestigationisleftforthefuture.
In proximity to g 1, severe fluctuations appear in the averaged phase of the Pfaffian – see
∼
Figure2–signalingthesignproblemmentionedintheDiscussion.
Acknowledgments
We are particularly grateful to M. Bruno for initial collaboration, and to R. Roiban, D. Schaich,
R. Sommer and A. Wipf for very useful elucidations. We also thank F. Di Renzo, H. Dorn, G.
Eruzzi, M. Hanada and the theory group at Yukawa Institute, B. Hoare, B. Lucini, J. Plefka, A.
Schwimmer,S.Theisen,P.Töpfer,A.TseytlinandU.Wolffforusefuldiscussions.
References
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6
LatticeandstringworldsheetinAdS/CFT:anumericalstudy ValentinaForini
350 70 20 6
300 60 5
250 50 15
numberofcnfgs 125000 numberofcnfgs 3400 numberofcnfgs 10 numberofcnfgs 234
100 20 5
50 10 1
0−1−0.8−0.6−0.4−0.2Re(0eiφ)0.2 0.4 0.6 0.8 1 0−1−0.8−0.6−0.4−0.2Re(0eiφ)0.2 0.4 0.6 0.8 1 0−1−0.8−0.6−0.4−0.2Re(0eiφ)0.2 0.4 0.6 0.8 1 0−1−0.8−0.6−0.4−0.2Re(0eiφ)0.2 0.4 0.6 0.8 1
Figure2: Histogramsforthefrequencyoftherealpart(theimaginarypartvanishesbysymmetry)
ofthereweightingphasefactoreiθ ofthePfaffianPfOF = (detOF)12 eiθ,basedontheensembles
generatedatg=30,10,5,1(fromlefttoright). Atg=1the| observed| eiθ isconsistentwithzero,
(cid:104) (cid:105)
thuspreventingtheuseofstandardreweighting.
g T/a L/a am Lm statistics[MDU] τ
int
×
10 16 8 0.5 4 900 2
×
20 10 0.4 4 900 3
×
24 12 0.33333 4 821,900 5
×
32 16 0.25 4 316,357 10
×
30 16 8 0.5 4 800 1
×
20 10 0.4 4 800 2
×
24 12 0.33333 4 900 4
×
32 16 0.25 4 625,800 8
×
40 20 0.2 4 300,300 60
×
100 16 8 0.5 4 900 1
×
20 10 0.4 4 750 2
×
32 16 0.25 4 411,415 5
×
Table 1: Parameters of the simulations. The temporal extent T is always twice the spatial extent
L, which helps studying the correlators. The size of the statistics after thermalization is given in
terms of Molecular Dynanic Units (MDU) which equal an HMC trajectory of length one. In the
caseofmultiplereplicathestatisticsforeachreplicaisgiven. Thetypicalauto-correlationtimeof
thecorrelatorsisgiveninthelastcolumn.
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