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Logarithmic corrections and universal amplitude ratios
in the 4-state Potts model
Bertrand Berche, Paolo Butera, Lev Shchur
To cite this version:
Bertrand Berche, Paolo Butera, Lev Shchur. Logarithmic corrections and universal amplitude ratios
in the 4-state Potts model. 2007. �hal-00164633�
hal-00164633, version 1 - 22 Jul 2007
B.Ber he 1 ,P.Butera 2 ,L.N.Sh hur 3 1Lab oratoire dePhysique desMateriaux,UniversiteHenriPoin are,Nan yI,
BP239,F-54506Vanduvre lesNan y Cedex,Fran e
ber helpm.u-nan y.fr 2
IstitutoNazionaledi Fisi aNu leare,UniversitaMilano-Bi o a, Piazzadelle
S ienze 3,20126,Milano,Italia
paolo.buteramib.infn.it 3
LandauInstitute forTheoreti alPhysi s,Russian A ademyofS ien es,
Chernogolovka 142432,Russia
levlandau.a .ru
Summary. Monte Carloand series expansion data forthe energy, sp e i heat,
magnetisation and sus eptibili ty ofthe 4-statePotts mo del in thevi inity ofthe
riti al p oint areanalysed. Therole of logarithmi orre tions is dis ussed.
Esti-matesofuniversal ratios A+=A , += L,
T = L andR + aregiven. July23, 2007 1 Introdu tion
Thestudyof riti alphenomenaandphasetransitionsisatraditionalsubje t
of statisti alphysi s whi h hasknownits \mo dernage", sin e p owerful
ap-proa heshaveb eendevelop ed(renormalizationgroup, onformalinvarian e,
sophisti ated simulationalgorithms, ...). Simpli edmo delsattra ted alot
of attention. Thisis essentially due to asp e ta ular prop ertyof ontinuous
phase transitions at their riti al p oint,s ale invarian e, whi h leads to an
extreme robustnessof somequantities,like the riti alexp onentswhi h are
thusreferred to as universalquantities.Only very general prop erties (spa e
or spin dimension,symmetry,range of intera tion, ...) determine the
uni-versality lass. This makesthe theoryof riti al phenomenaaveryeÆ ient
and predi tive to ol: As so on as one knows the general hara teristi s of a
physi alsystemfromgeneral symmetryarguments,itisp ossibleinprin iple
topredi texa tlythe\shap e"ofthesingularitieswhi haredevelop edatthe
riti al p oint.The term\exa t" is here understo o d rigorously,for example
atwo-dimensionalsystem withthesymmetries ofan Ising mo del,should it
b eamagnet,analloyoranythingelse,willexhibitadivergingsus eptibility
jT T
j
7=4
(see asket h inFig.1) withthe pre isevalue7=4 forthe
exp onent.
On a theoreti al ground, the major two-dimensional problems (Ising
0.7
0.8
0.9
1
1.1
1.2
1.3
T / T
c
0
30
60
90
120
150
χ = Γ
+
τ
−γ
χ = Γ
−
|τ|
−γ
’
τ > 0
τ < 0
0
Fig.1. Typi alb ehaviourofthesus eptibili tyatase ondorderphasetransition.
Thequantities =
0
and += areuniversal.
XY mo del,Heisenb erg mo del and so on)are essentially solved at least for
their riti al singularities (when a se ond-order phase transition is indeed
present), but riti al exp onents are not the only universal quantities at a
riti al p oint. The universal hara ter of appropriate ombinations of
rit-i al amplitudes[1℄ is also an imp ortant predi tion of s aling theory but in
some asesthese ombinationsremainun ompletelydeterminedandsubje t
to ontroversies.
The Potts mo del [2,3℄, as one of the paradigmati mo dels exhibiting
ontinuous phase transitions is a go o d frame to onsider the question of
universal ombinations of amplitudes. The universality lass of the Potts
mo del at its riti al p oint is parametrized by the numb er of states q . The
two-dimensionalPotts mo delwith three and four states an b e exp
erimen-tally realized as strongly hemisorb ed atomi adsorbates on metalli
sur-fa esatsub-monolayer on entrations[4℄.Although riti alexp onents ould
b e measuredquitea urately foradsorb ed sub-monolayers, on rmingthat
thesesystemsa tuallyb elongtothethree-state[5℄ortothefour-statePotts
mo del lasses [6℄,it isunlikelythat the lowtemp erature LEED results an
b e pushed [7℄ to determine also the riti al amplitudes. Therefore, the
nu-meri alanalysis of these mo delsis the onlyavailableto olto he k analyti
predi tions.
The riti alamplitudesand riti alexp onents des rib ethe b ehaviour of
themagnetizationm,thesus eptibility ,thesp e i heatC andthe
orre-lationlength foraspinsysteminzeroexternal eld
4
inthevi inityofthe
riti alp oint
M( )B ( ) ; <0; (1) ( ) j j ; (2) T ( ) T ( ) ; <0; (3) C( ) A j j ; (4) ( ) 0 j j : (5)
Here istheredu ed temp erature =(T T
)=T andthelab elsrefer to
thehigh-temp erature and low-temp eraturesides ofthe riti al temp erature
T
. For the Potts mo delswith q >2 atransverse sus eptibility
T
an b e
de ned inthelow-temp eraturephase
5 .
Criti alexp onentsareknown exa tlyfor2DPottsmo del[8{11℄through
therelationx
=(1 )= tothethermals aling dimension
x = 1+y 2 y (6)
andtherelationx
= = to themagneti s alingdimension
x = 1 y 2 4(2 y ) ; (7)
where the parameter y is related to the numb er of states q of the Potts
variablebytheexpression
os y 2 = 1 2 p q (8)
The entral hargeofthe orresp onding onformal eldtheoryisalsosimply
expressed [9℄intermsofy =1 3y 2 2 y : (9)
Analyti alestimatesof riti alamplituderatiosfortheq -statePottsmo
d-elswithq=1,2,3,and4werere entlyobtainedbyDel noandCardy[12℄.
Theyusedthetwo-dimensionals attering eldtheoryofChimandZamolo
d- hikov[13℄andestimatedthe entral harge =0:985for4-statePottsmo del,
forwhi htheexa tlyknownvalueis =1.Rep ortingtheseapproximate
val-uesin(9), one an al ulatethes alingdimensionsfrom(6)-(7)andgetthe
valuesx
=0:13016andx
=0:577,tob e omparedresp e tivelytotheexa t
values1=8and1=2.Thedis repan yisaround4and15p er ent,
emphasiz-ingthediÆ ultyoftheq=4 ase (togiveanidea,inthe aseofthe3-state
Pottsmo del,asimilaranalysisleadstoaverygo o dagreementwithlessthan
one p er ent deviation).
Theuniversalsus eptibilityamplituderatios
+ = L and T = L werealso
al ulatedin[12℄and[14℄.The gures obtainedarethefollowing,
5
Inthefollowing wewill useequallythenotations
L
q=3: + = L =13:848; T = L =0:327; (10) q=4: + = L =4:013; T = L =0:129: (11)
These results have b een on rmed numeri ally inthe ase q=3by several
groups, + = L 10and T = L
0:333(7)inRef.[14℄(MonteCarlo(MC)
simulations), +
= L
=141inRef.[15℄(MCandseriesexpansion(SE)data)
andquitere ently, theseresultswere on rmedandsubstantiallyimproved,
+ = L = 13:83(9), T = L
= 0:325(2), by Enting and Guttmann [16℄who
analysednewlongerseriesexpansions.
The 4-state Potts mo del was also studied through MC simulations in
Ref.[14℄,buttheauthors onsideredthattheirdatawerenot on lusive.
An-other MC ontribution isrep orted byCaselle,et al[17℄,
+ =
L
=3:14(70),
and Enting andGuttmann[16℄also analysedSE data for the4-state Potts
mo delandfound
+ = L =3:5(4), T = L =0:11(4)inrelativelygo o d
agree-mentwith the predi tions of [12℄and [14℄. The situationthus seems to b e
lear,althoughtheuseofthelogarithmi orre tionsinthe ttingpro edure
ofMCdatawasquestioned,e.g.in[16℄:[Caselleetal℄estimatesdepend
rit-i ally on the assumed form of the sub-dominant terms, and on the further
assumptionthattheother sub-dominantterms,whi hin ludepowersof
loga-rithms,powersoflogarithmsoflogarithmset , anallbenegle ted.Wedoubt
that thisis true.
Let us re all that the existen e of logarithmi orre tions to s aling in
the 4-state Potts mo del was p ointedout inthe pioneering works ofCardy,
Nauenb erg and S alapino[18,19℄,where a set of non-linear RG equations
wereprop osed.Theirdis ussionwaslaterextendedbySalasandSokal[20℄.
Generi ally,the logarithmi orre tions app ear as orre tions tos aling.
We mentionedab ove that inthevi inityof a riti alp oint,asus eptibility
for example diverges like ( )
j j
. This is true, but this singular
b ehaviour anb esup erimp osedtoaregularsignal(e.g.D
0 +D
1
j j+:::),and
theleadingsingularb ehaviouritselfneeds tob e orre tedwhenwe onsider
thephysi alquantityawayfromthetransitiontemp erature. Theexpression
for ( )thentakes aformwhi h anb e ome\terri ":
( )=D 0 +D 1 j j+::: regularba kground + j j (1+ leadingsingularity +a (1) j j +a (2) j j 2 +::: leading orre tions +a 0 (1) j j 0 +a 0 (2) j j 2 0 +:::next orre tions +b (1) j j+b (2) j j 2 +:::) analyti orre tions ( lnj j) ? 1+/ ln( lnj j) lnj j ::: logarithmi orre tions
Togetherwith theamplitudeand exp onent asso iatedto theleading
singu-larity, and ,app ear orre tionstos alingduetothepresen eofirrelevant
s aling elds(a (n) and,a 0 (n) and 0
,...),analyti orre tionsdueto non-(n)
orre tions (/ and ?,...).These logarithmi o eÆ ientsmayhavedi erent
origins(see e.g.inRef.[1℄ andreferen es therein). They an b e due to the
upp er riti al dimension,to p oles inthe expansion of regular and singular
amplitudes, or to thepresen e of marginals aling elds. The4-state Potts
mo delb elongsto thislatter ategory.
Someofthe quantities indi atedab ove are universal.This isthe ase of
theexp onentsaswellasofmany ombinationsof o eÆ ients.Inthepresent
pap erweareinterestedintheamplitudeoftheleadingsingularterm,butits
pre isedetermination anb ea e tedbytheformofthe orre tions.Weshall
b e on ernedwiththefollowinguniversal ombinationsof riti alamplitudes
A + A ; + L ; T L ; R + C = A + + B 2 : (12)
We present, for the 4-state Potts mo del, more a urate Monte Carlo data
supplementedbyareanalysisoftheextendedseriesmadeavailablebyEnting
and Guttmann[16℄and we address thefollowingquestion: Isit p ossible to
devise somepro edure in whi h therole of these logarithmi orre tions is
prop erlytakenintoa ount?
2 Amplitudesand universal ombinations
Thes alinghyp othesisstatesthatthesingularpartofthefreeenergydensity
anb ewrittenintermsofthedeviationfromthe riti alp oint, =(T T
)=T andh=H H , f sing ( ;h)=b D F ( b y ; h b yh h) (13) whereF
(x;y )isauniversalfun tion(a tuallythereisoneuniversalfun tion
forea hside >0or <0ofthe riti alp oint)and
and
h
are\metri
fa tors"whi h ontainallthenonuniversalasp e tsofthe riti alb ehaviour.
Disthespa edimension.Letusstressthatthefun tionsF
areuniversalin
thesensethatsomedetailsofthemo delareirrelevant(e.g.the o ordination
numb erofthelatti e(solongasitremains nite),thepresen eofnextnearest
neighb ourintera tions,et ) but they dep end ontheb oundary onditionsor
the shap e of the system. The metri fa tors on the other hand dep end on
these details,and theuniversal ombinationsareobtainedwhen themetri
fa torsareeliminatedfromsome ombinations.
The onne tion with s aling relations an b e shown with an
exam-ple. From Eq. (13), we also dedu e similar homogeneous expressions for
the magnetization, M( ;h) = b D +y h h M
(x;y ) and the sus eptibility,
( ;h) =b D +2y h 2 h X (x;y ).The hoi e b=( j j) 1=y and h=0leads
(forexampleb elowthetransitiontemp erature)forthefollowing ombination
C( ;0) ( ;0) m 2 ( ;0) j j 2 ( j j) 2 2 C (1;0)X (1;0) M 2 (1;0) R: (14)
The prefa tor takes the value 1 thanks to the well known s aling
rela-tion b etween riti al exp onents +2 + = 2. Thus it follows that the
ab ove ombinationisauniversalnumber.Fromthede nitionof
magnetiza-tion, sp e i heat and sus eptibility amplitudesin zero magneti eld,e.g.
M( ;0) =j j D y h y h M (1;0) B j j
by virtueof Eq. (1), this universal
numb er is in fa t a ombination of amplitudes, R A =B
2
. We have
similaruniversal ombinationsab ove the riti altemp erature or asso iated
toother s alingrelations.
3 RG approa h for the Potts model and logarithmi
orre tions at q =4
Letusremindthattheq -statePottsmo delisanextensionoftheusuallatti e
Ising mo delin whi h thesite variabless
i
(abusively alled spins) an have
q di erent values,s
i
=0;1;::: q 1but the nearest neighb ourintera tion
energy JÆ
s i
;s j
only takes two p ossible values, e.g. J and 0 dep ending
whethertheneighb ouringspinsareinthesamestateornot.TheHamiltonian
ofthemo delreadsas
H= J X hiji Æ s i s j : (15)
Attheearly timesofreal-spa erenormalization,theappli ationto thepure
Potts mo delled to somediÆ ulties: theimp ossibilityto a e t a parti ular
valuefor the spin of a ell after de imation due to a to o large numb er of
states (see Fig.2).
_
_
_
_
_
_
_
+
+
+
+
+
+
_
_
+
0 6
4
5
5
1
3
2
6
6
6
7 7
7
7
0
0
0
0
3
5
5
5
4
4
2
2
6
? ?
+ +
+
+ +
+
_ _
_
_
_
_
_
+ +
+
+
+
+
+
1
1
1
1 1
1
Fig.2. De imationofspinblo ksfortheIsingmo del(left)andhigh-qPottsmo del
(right).Inthe latter ase,the state ofmany ells annot b e de ided by asimple
majorityrule.
to study anannealed disordered mo del.The RGequations satis ed by the
mo del,writtenintermsoftherelevantthermalandmagneti elds andh,
with orresp ondingRGeigenvaluesy
andy
h
,andthemarginaldilution eld
, aregivenby d dlnb =y , dh dlnb =y h h, d dlnb =q q
,whereb isthelength
res aling fa tor and l = lnb.When q >q
, the dilution eld is relevant
(and the phase transition is of rst order), while in the regime q < q
,
is irrelevant and the system exhibits a se ond-order phase transition. The
ase q=q
is marginal.This pi tureis qualitatively orre t,and infa t the
riti al value of the numb er of states whi h dis riminates b etween the two
regimes is q
=4.In theq dire tion,q
=4app ears as theend of aline of
xed p ointswhere logarithmi orre tions are exp e ted. Atq
=4,theRG
equationswereextendedbyCardy, Nauenb erg andS alapino(CNS)[18,19℄
and then bySalas and Sokal(SS) [20℄.As aresult of the oupling b etween
thedilution eld ,and andh,they wereled tonon-linearequations,
d dlnb =(y +y ) ; (16) dh dlnb =(y h +y h )h; (17) d dlnb =g ( ): (18)
Thefun tiong ( ) mayb e Taylorexpanded, g ( )=y
2 2 (1+ y 3 y 2 +:::).
A ounting for marginalityof the dilution eld, there is no linear term at
q=4.Comparingto theavailableresults (forexampletheexpressionofthe
latentheatforqq
byBaxter[21℄orthedenNijsandPearson's onje tures
fortheRGeigenvaluesforqq
[10,11℄),theparameterswerefoundtotake
the values y = 3=(4 ),y h = 1=(16 ),y 2 = 1= and y 3 = 1=(2 2 ),
whiletherelevants aling dimensionsarey
= 1 =3=2andy h =15=8.
The xed p ointis at = h = 0. Starting from initial onditions , h,
therelevant elds growexp onentiallywithl upto some =O (1),h=O (1)
outside the riti al region.Noti e alsothat the marginal eld remains of
orderofitsinitialvalue, O (
0
).Inzeromagneti eld,undera hangeof
lengths ale,thesingularpartofthefreeenergydensitytransformsa ording
to f( 0 ; )=e D l f( ;1): (19)
SolvingEqs.(16-18)leads to
l= 1 y ln + y y y 2 ln 0 G( 0 ; ) ; (20)
(forbrevitywewilldenote=1=y
= 2 3 ,= y y y 2 = 1 2 ).NotethatG( 0 ; )
wouldtakethevalue1inRef.[19℄andthevalue
y 2+y 3 y 2 +y 3 0 inRef.[20℄.We
f( ; 0 )= D 0 y 2 +y 3 0 y 2+y 3 D f(1; ): (21)
A similar expression would b e obtained if the magneti eld h were also
in luded. Theotherthermo dynami prop erties followfromderivatives with
resp e t to the s aling elds. The quantity b etween parentheses is the only
one wherethelogtermsarehiddeninthe4-statePotts mo del,andthus we
mayinferthatnotonlytheleading logterms,butallthe logtermshidden in
the dependen e on the marginal dilution elddisapp ear inthe onveniently
de nede e tiveratios
6
.Nowwe pro eedbyiterationsofEq.(20),and
even-tuallywe getforthefull orre tion to s alingvariabletheheavyexpression
0 G( 0 ; )= onst( lnj j) | {z } CNS 1+ 3 4 ln( lnj j) lnj j 1 3 4 ln( lnj j) lnj j 1 | {z } 1+ 3 2 ln( lnj j) lnj j inSS 1+ 3 4 1 ( lnj j) 1+ onst lnj j +O 1 lnj j 2 | {z } F( lnj j) (22)
where CNS and SS refer to the results previously obtained in the
litera-ture [18{20℄and F( lnj j)istheonlyfa tor wherenon universalityenters
throughthedilution eld
0
.Thisallowstowritedowntheb ehaviourofthe
magnetizationforexample
M( )=B j j 1=12 ( lnj j) 1=8 1+ 3 4 ln( lnj j) lnj j 1 3 4 ln( lnj j) lnj j 1 1+ 3 4 1 lnj j F( lnj j) # 1=8 : (23) 4 Numeri al te hniques
IntheMonte Carlosimulationswe usetheWol algorithm[22℄forstudying
square latti es of linear size L (b etween L = 20 and L = 200) with p
eri-o di b oundary onditions.Startingfromanorderedstate,we letthesystem
equilibrate in 10
5
steps measuredby the numb er of ipp ed Wol lusters.
The averagesare omputed over 10
6 |10
7
steps. Therandom numb ers are
pro du ed byan ex lusive-XOR ombinationof twoshift-registergenerators
with the taps (9689,471) and(4423,1393),whi h are known [23℄to b e safe
fortheWol algorithm.
Theorderparameterofami rostateM(t) isevaluatedduringthe
simula-tionsas
M= m q 1 ; (24) whereN m
isthenumb erofsitesiwiths i
=matthetimetofthesimulation
andm2[0;1;:::;(q 1)℄isthespinvalueofthemajorityofthesites.N =L
2
isthetotalnumb erofspins.ThethermalaverageisdenotedM =hMi.Thus,
thelongitudinalsus eptibilityinthelow-temp eraturephase ismeasuredby
the u tuationofthemajorityofthespins
L = (hN 2 m i hN m i 2 ) (25)
andthe transverse sus eptibilityis de nedinthe low-temp eraturephase as
the u tuationsof theminorityofthespins
T = (q 1) X 6=m (hN 2 i hN i 2 ); (26)
while inthehigh-temp erature phase
+
is givenbythe u tuations inallq
states, + = q q 1 X =0 (hN 2 i hN i 2 ); (27) where N
is the numb er of sites with the spinin thestate . Theinternal
energy densityof ami rostateis al ulatedas
E= 1 N X hiji Æ s i s j (28)
its ensemble average denoted as E = hEi and the sp e i heat p er spin
measurestheenergy u tuations,
C= 2 E = 2 hE 2 i hEi 2 : (29)
Our MC study of the riti al amplitudes is supplemented by an
anal-ysis of the high-temp erature (HT) and low-temp erature (LT) expansions
for q = 4 re ently al ulated through remarkably high orders by Enting,
Guttmannand oworkers [16,24℄.In termsof these series, we an ompute
the e e tive riti al amplitudes for the sus eptibilities and the
magnetiza-tion and extrap olatethem bythe urrent resummationte hniques, namely
simplePadeapproximants(PA)anddi erentialapproximants(DA)prop erly
biasedwith theexa tly known riti altemp eraturesand riti alexp onents.
TheLTexpansions,expressed intermsofthevariablez=exp( ), extend
throughz
59
forthelongitudinalsus eptibilityandthroughz
47
inthe ase of
the transverse sus eptibility. The magnetizationand energy expansions
ex-tend through z
43
. The HT expansion is omputed in termsof thevariable
v=(1 z)=(1+(q 1)z).Thesus eptibilityexpansionhasb een omputed
5 Analysis of the magnetizationbehaviour
For thesake ofsimpli ation ofthe notations, we group allthe terms
on-taining logs in Eq. (22) into a single fun tion H( lnj j) = E( lnj j)
F( lnj j)where E( lnj j)= ( lnj j) 1+ 3 4 ln( lnj j) lnj j 1 3 4 ln( lnj j) lnj j 1 1+ 3 4 1 lnj j # : (30)
Thefun tionE ontainsallleadinglogarithmswithuniversal o eÆ ients,it
is known exa tly while the fun tion F needs to b e tted. We thus obtain
a losedexpression for the dominantlogarithmi orre tions whi h is more
suitablethanpreviouslyprop osed formsto des rib eanobservable(Obs.) in
thetemp eraturerangea essibleinanumeri alstudy:
Obs:( )'Ampl:j j
H
℄
( lnj j)(1+Corr:terms); (31)
Corr:terms=aj j
2=3
+b
j j+:::; (32)
whereand℄areexp onentswhi hdep endontheobservable onsidered,and
takethevalues1=12and 1=8resp e tivelyinthe aseofthemagnetization.
Herewestressthatthein lusionofa orre tioninj j
2=3
seemstob ene essary
a ording to previous work of Joy e on the Baxter-Wu mo del [25,26℄ (of
4-state Potts mo del universality lass), where the magnetization is shown
to ob ey an expression of the form M( ) = B j j
1=12 (1+ onstj j 2=3 + onst 0 j j 4=3
). The exp onent 2=3 omes out from the onformal s aling
dimensionsofDotsenkoandFateev[9℄,anditspresen eisneededinorderto
a ountforthenumeri alresults (see alsoRef.[27℄).Caselle et al.[17℄also
onsideredj j 2=3
termto tthemagnetization.Here wealso allowin lusion
of a linear orre tion in b
j j to a ount for p ossible nonlinearities of the
relevants aling elds[1℄.Thenextterminj j
4=3
willb eforgotten.
In Fig. 3 we plot e e tive magnetization amplitudes B
eff
( ) vs j j 2=3
.
Fromtheavailabledataforthemagnetization,wede nethefollowing
quan-tities, B CNS ( )=M( )j j 1=12 ( lnj j) 1=8 ; (33) B SS ( )=M( )j j 1=12 ( lnj j) 1=8 1 3 16 ln( lnj j) lnj j 1 ; (34) B E ( )=M( )j j 1=12 E 1=8 ( lnj j); (35)
whi hareexp e tedtob ehave a ording tothe orre tionsto s aling
B (1+aj j
2=3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
|τ|
2/3
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
B
eff
(
τ
)
no log
CNS
SS
E(- ln |
τ|
)
1 + a|
τ
|
2/3
1 + a|
τ|
2/3
+
b|
τ
|
region of fit including a and b coefficients
Fig.3. E e tiveamplitudes(fromMCdata)asdedu edfromdi erentassumptions
forthelogarithmi orre tions.Symb ols orresp ondtoMCdata,dashedand
dotted-dashedlines are tsasexplained inthetext.
Thenumeri alresults areshown inFig.3.Fromb ottomto topthevarious
symb ols indi ate the e e tive amplitudes with "no-log" at all, then with
the CNS and the SS orre tions and nally the e e tive amplitude where
theknownuniversallogarithmi termshaveb eenin luded.Thedashedlines
orresp ond to arough determinationofthe orre tions to s aling in luding
only the terms in aj j
2=3
in the limitj j ! 0, and the dot-dashed lines
in lude also the terms in bj j.From this plot, we dedu e that none of the
threee e tive amplitudes inEqs.(33-35) an be orre tly ttedby Eq.(36),
sin e the o eÆ ients of the orre tion terms (e.g. the o eÆ ient a whi h is
estimated dire tly by the slop eat smallj j values) stronglydep end on the
rangeof t.This undesirabledep enden e of the o eÆ ients onthewidth of
thetemp eraturewindowisshowninthe rstsixlinesoftable1.
In order to improve the quality of the ts, one has to take into
a - ountthe orre tionfun tionF( lnj j)andtoextra tane e tive fun tion
F eff
( lnj j)whi hmimi stherealoneinthe onvenienttemp eraturerange.
Thisisdoneby ttingB
eff
( )toamore ompli atedexpression,
B (1+aj j 2=3 +b j j+:::) 1+ C 1 lnj j + C 2 ln( lnj j) ( lnj j) 2 1=8 ; (37)
whi hmeansthatwein ludethe orre tionstos alingandthenonuniversal
fun tionfun tionF( lnj j)takingtheapproximateexpression
F eff ( lnj j)' 1+ C 1 + C 2 ln( lnj j) 2 1 : (38)
Table 1.Fits ofthe e e tiveamplitudeofthemagnetization. Beff( ) j j 2=3 -window B a b BCNS( ) [0;0:15℄ 1:07 0:98 0:94 [0;0:45℄ 1:05 0:66 0:29 B S S ( ) [0;0:15℄ 1:11 0:77 0:56 [0;0:45℄ 1:10 0:45 0:06 B E ( ) [0;0:15℄ 1:14 0:47 0:16 [0;0:45℄ 1:13 0:25 0:27 B E F ( ) [0;0:15℄ 1:16 0:20 0:02 [0;0:45℄ 1:16 0:18 0:02
Whileaandbare o eÆ ientsof orre tionstos alingduetoirrelevantop
era-tors,C
1
andC
2
aree e tive o eÆ ientsoflogarithmi termswhi h,inagiven
temp eraturerange,mimi aslowly onvergentseriesoflogarithmi terms
de-p endingonanonuniversaldilution eld.Therefore,we exp e tthatdi erent
ts made indi erent temp erature windowswillpro du e di erent values of
C 1
and C
2
while a and b (and of ourse also the magnetizationamplitude
B )should b erelativelylessin uen ed bythewindowrange.The hoi e of
valuesforC
1
andC
2
isthuspartiallyarbitraryandthevaluesquotedshould
b esp e i edtogetherwiththetemp eraturewindowwherethey are
appropri-ate.In thefollowing,weobtainC
1 ' 0:76and C 2 ' 0:52inthewindow j j 2=3
2[0;0:35℄,whi hyieldsanamplitudeB '1:157.Theresultingaand
b o eÆ ients now app ear very stable. This is he ked in Fig. 4 where the
quantity B EF ( )=M( )j j 1=12 [E( lnj j)F( lnj j)℄ 1=8 (39)
is rep orted toghether with the previous urves and tted as indi ated in
table1.Asanindep endenttest,we addtheSEdatawhi haresup erimp osed
to the MC data at small values of j j only for this latter assumption of
e e tive amplitude.
Eventually,ourapproa h on rmsexpressionEq.(23)forthe
magnetiza-tion, withthefun tionF( lnj j)giveninEq. (38)andtheparameters C
1
and C
2
given ab ove forthe appropriatetemp erature window.Nevertheless,
wehave to stress that thedi erente e tive amplitudesshouldallrea h the
sameamplitudeB inthelimitj j!0,sin eB
CNS ( )isinfa tan approx-imationof B SS ( ), whi h is an approximationof B E ( ), whi h eventually approximatesB EF
( ).Anattemptofillustrationofthisb ehaviourisshown
inFig.5wheredottedlines(whi hareonlyguidesfortheeyes)all onverge
towardsthe uniquevalueB '1:157.Here we stress that we have deleted
thep ointsofSEdatawhi hareto o losetothe riti alp oint,sin etheseries
are no longer reliable b e ause the urrent extrap olation pro edures are in
prin ipleunabletoapproximatethe ompli atedstru ture ofthesingularity
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
|τ|
2/3
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
B
eff
(
τ
)
no log
CNS
SS
E(- ln |
τ|
)
E(- ln|
τ
|)
x F(- ln|
τ|
)
1 + a|
τ
|
2/3
1 + a|
τ|
2/3
+
b|
τ
|
region of fit including a and b coefficients
Fig.4. E e tiveamplitudes(fromMCandSEdata)asdedu edfromdi erent
as-sumptionsforthelogarithmi orre tions.Theupp er urves orresp ondtoB
E F ( )
andthethi ksolid linestoSEdata.
0
0.05
0.1
|τ|
2/3
1
1.05
1.1
1.15
B
eff
(
τ
)
B
CNS
B
SS
B
E
B
EF
guide for the eyes
1 + a|
τ|
2/3
+
b|
τ
|
Fig.5. Zo omofthee e tive amplitudesin thevi inity of the riti al p oint. MC
data(symb ols)andSEdata(thi ksolid lines).
6 Universal ombinations for the 4-state Potts model
and on lusions
The other quantities an b e analyzed along the samelines. The imp ortant
era-tures aleandthustheremainingfreedomfortheotherphysi alquantitiesin
Eq. (31)is onlythrough theleadingamplitudeandthe o eÆ ientsofj j
2=3
andj j-termsinthe orre tionsplusp ossiblytheba kgroundterms.Itisstill
a ompli ated task to p erformthis analysis,but the urrent results for the
universal ombinationsmentionedintheintro du tionapp earinthefollowing
table.
Table 2. Roughestimate oftheuniversal ombinations ofthe riti al amplitudes
inthe4-statePottsmo del.
A + =A + = L T = L R + C sour e 1: a 4:013 0:129 0:0204 [12,14℄ 3:14(70) 0:021(5) [17℄ 3:5(4) 0:11(4) [16℄ 1:00(1) 6:7(4) 0:161(3) 0:0307(2) here a
exa tresultfromduality
Our work is "one more" ontribution to the study of this problem and
brings someanswers, but also raisesnewquestions. Indeed,ourresults
dis-agreewithpreviousestimates,butwe annot laimforsurethatourestimates
aremorereliablethanthoseofotherauthors.Whatisextremely learisthat
thegroupswhostudied numeri allyuniversal ombinationsofamplitudesin
the4 statePottsmo delallnoti edtheextremediÆ ultytotakeintoa ount
prop erlythelogarithmi terms.Web elievethatourproto olisself- onsistent
inthesense that our riterionis to obtainarelative stabilityofthe
orre -tiontos aling o eÆ ients.Theresultsthatwerep orthere,althougharough
estimatewhi h allsfordeep er analysis,rea h areasonable on den elevel.
If this is indeedthe ase,one should identifythe reason of the dis repan y
fromthetheoreti alpredi tionsofCardyandDel no.Inthe on lusion,and
in afo otnote of one of their pap ers, Del no et al[14℄ (p.533)explain that
theirresults aresensitive totherelativenormalizationof theorderand
dis-order op erator formfa tors whi h ould b e the origin of sometroubles for
the ratios + = L andR C
. This p ossible explanationseems nevertheless to
b e ruledout (as mentionedby Enting and Guttmannalready) by the very
go o dagreementb etweenthetheoreti alpredi tionsandallnumeri alstudies
(b othMC andSE) inthe ase ofthe3-state Potts mo del.Eventuallyletus
mentionthatthetwo-kinkapproximationusedbyCardyandDel noisexa t
forq=2(Isingmo del)and quitego o dforq=3,but probablyquestionable
lose to the marginal ase q ! 4. As a on lusion, we are afraid that this
A knowledgements
Dis ussions with A. Zamolo d hikov,V. Ple hko, W. Janke and M. Henkel,
anda orresp onden e withJ.CardyandJ.Salaswerevery helpful.
LNSis grateful to the Statisti al Physi s group of the UniversityHenri
Poin are Nan y 1 and to the Theoreti al group of the University Milano{
Bi o a for the kind hospitality. Finan ial supp ort fromthe twin resear h
programb etween theLandauInstituteandtheE oleNormaleSuperieurede
ParisandRussianFoundationforBasi Resear h arealsoa knowledged.
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