Emergene of Log-Periodi Osillations in
Periodi and Aperiodi Ising Models
R.F. S. Andrade
InstitutodeF
isia-UniversidadeFederalda Bahia
CampusdaFedera~ao
40.210-340 -Salvador -Brazil
Reeivedon15August,2000
Thisworkanalyzestheemergeneoflog-periodiosillationsinthermodynamifuntionsofIsing
models on hierarhial latties. Several situations, where the exhange interations are periodi
or aperiodi, are taken into aount. Highpreision valuesfor the thermodynamifuntions are
numeriallyobtainedwiththemethodoftransfermatries. Fittingtheurveslose totheritial
temperature leadstothe valuesofthe ritial exponentsand to theperiodand amplitudeofthe
osillations. Thersttwoquantitiesarefoundtoagreewiththeresultspreditedbythe
renormal-izationgroup. Theamplitudeofosillations,whihareminuteforbothperiodisystemsandthose
withaperiodiirrelevantutuations,aresigniantlyenhanedforsystemswithaperiodirelevant
utuations. Distint morphologiesofthe osillating patternaredisussed, whereosillations are
respetivelysinusoidalandwithasigniantontributionofhigherorderharmonis.
I Introdution
Log-periodi osillationsonstitute ageneralproperty
ofsystemswithdisretesaleinvariane[1℄. Theyhave
beenreportedinmanydierentsystemsinwhihthere
is disrete sale invarianein the lattie onwhih the
model is dened (e.g. geometri fratals [2, 3℄), orin
the way oupling onstants appear over a Eulidean
lattie[4℄, orboth[5,6℄. Theirpresene hasbeen
dis-ussedsinetheearlydaysoftherenormalizationgroup
(RG) approahtophasetransitionsin spinmodels[7℄,
as a multipliative log-periodi fator in terms of the
reduedtemperatureappearsinthegeneralsolutionof
therenormalizationtransformation. Despitethis, only
reently has this phenomenon reeived due attention,
in the form of detailed analyses of some models and
reviews.
Thisanbeunderstoodbythefatthatthese
osil-lationsgenerallyhavesuhminuteamplitudesthatthey
arehardlydetetedinthemostusualanalyses,e.g.,in
the evaluation of theritial exponents. However,the
reent investigation of a large number of
determinis-tiaperiodimodels,whereutuationsinthevalueof
the ouplingonstantsalterthesystem fromthe
orig-inal periodi ounterpart, has brought this subjetto
light. Thereasonanbebetterdisussedinthelightof
Luk's riterion for relevant or irrelevant utuations
[8℄. Aording to it, relevant utuations are able to
promoteastruturalhangeintheRGowdiagramof
the model, sothat the properties at the ritial point
are nolongerdesribedbythexed point(FP)of the
evantutuationsleaveunhangedtheowdiagram,so
that the ritial exponents remain the sameas in the
homogeneousase. Therelevantorirrelevantharater
dependsonthespinmodel,onthelattieused,andon
thegeometripropertiesoftheaperiodisequenethat
is used to dene the new values of the oupling
on-stants. The important feature herein is the fat that
relevantutuationshaveageneraltendenytoweaken
ritiality. Sotheyenhane the magnitudeof the
log-periodiosillationsoftheaperiodisystemswhihan
beidentiedandonvenientlyanalyzed.
The above observations will be illustrated in this
work,whih analyzeslog- periodiosillationsin
peri-odiandaperiodiIsingmodelsonhierarhiallatties.
Weproeed with theevaluation ofall thermodynami
funtionsofthemodelswithatransfermatries(TM)
sheme[9℄,whihamountstoiteratingaseriesofmaps
forsuhfuntionsatthesuessivestepsofonstrution
of the lattieuntil numerial onvergene is ahieved.
Thiswork isthethird ofaseriesof papers;in the
for-mer two [5, 6℄, we developed the method for the ase
ofhierarhiallatties,investigatedthegeneral
proper-ties of some models and haraterizedthe osillations
forasituationwithaperiodirelevantutuations. We
onentrate now in theharaterizationof osillations
fortwootherases,onewithperiodiinterations,the
other with aperiodi irrelevant utuations. We show
that,despitemuhsmalleramplitudes thanin the
for-merase,osillationsarepresentforallthermodynami
funtions.Theirperiodanbeexpressedintermsofthe
morphology, we show that, for the periodi ases,
os-illations are almost sinusoidal for both T > T
and
T < T
. On the other hand, for the irrelevant
aperi-odi ase,osillations havesigniantontributionsof
at leasttwoharmonisfor T belowand aboveT
. In
bothases,themorphologyoftheosillationsisdistint
fromthosein themodelwithrelevantutuations.
Therestoftheworkisorganizedasfollows: InSe.
II wedene the hierarhial lattiesand aperiodi
se-quenesusedtoindueutuationsinthespin-spin
ex-hange ouplings; in Se. III we disussthe key steps
ofthetransfermatrixformalism;inSe. IVwepresent
theresultsobtainedfromttingthedata,wherewe
em-phasizethequestionofsingleorhigherorderharmoni
ontributions in the periodi funtion; nally, Se. V
loseswithonludingremarks.
II Hierarhial latties and
ape-riodi sequenes
Hierarhiallatties [10, 11, 12℄ are graphswhih are
onstrutedbystartingfromasinglelinesegment
link-ing two (root) sites and following a substitution rule
thatamounts,forallsubsequentgenerations,to
repla-ing any single segment by a unit ell omposed of a
set of line segmentsand intermediate sites. Their
im-portane for the analysis of phase transitions of
mag-neti systems is mainly due to two fats: i) they are
exatly sale invariant, so that they are suitable for
theuse ofmathematial toolsbased onthis key
prop-erty, as RG and the adapted TM; ii) despite the fat
that they annotberealized in anynite dimensional
Eulidean spae, magneti models dened upon suh
lattiesonstitute approximations,within the
Migdal-Kadanof renormalization group, to the orresponding
modelsonEulideanlatties,sothattheymayleadto
resultswithsomerelevaneforatualphysialmodels.
Forthe presentwork weonentrateonmodels where
thebasiellisonstitutedbypparallelbranhes,eah
oneontainingaseriesofbsegments. Thefratalgraph
dimensionofsuhlattieisexpressedby
d
f =
log (pb)
log (b)
; (1)
sothat,ifwetakeb=p;wewillkeepd
f
=2onstant.
WedenetheformalHamiltonianforour
investiga-tionby
H = X
(i;j) J
ij
i
j h
X
i
i
; (2)
where
i
=1indiatesIsingspinvariables,thedouble
sum(i;j)isperformedoverpairsofrstneighborsites.
The bonds J
ij
are either onstantor dened withthe
help of the aperiodi sequenes whih wewill disuss
soon. Inthisase,theyassumethesamevaluesforany
G
ofonstrutionofthelattie. Finallytheeldhis
uni-formforallsitesofthelattie,independentofitsloal
oordination.
Inationorsubstitutionrulesof2ormoresymbols
areaonvenientwayto generatesequeneswhihmay
beperiodioraperiodi. Weonsider hereintwo
sym-bol(AandB)inationruleswhiharedesribedby
(A;B)!(AB b 1
;A b
) (3)
where the sequene starts with one single A symbol.
With exeption of thetrivial b = 1ase, all other
se-quenesareknowntobeaperiodi. Thesesequenesare
veryonvenienttogenerateaperiodimodelsonthe
hi-erarhiallattiesjustdisussed. Forthispurposeitis
importanttohoosethesamevalueforbin(1)and(3),
insuhawaythatthelengthofthepathsbetweenthe
rootsites ofthelattiematheswith thelengthof the
sequene for any generation G.So, hoosing J
ij to be
eitherJ
A orJ
B
;aordingtotheorrespondingsymbol
in the aperiodi sequene, warrantsthat an aperiodi
modelwillbegeneratedin thelattie.
Theutuations in the exhange onstantshavea
relevant (or irrelevant) harater, aordingto Luk's
riterion,adaptedtohierarhiallatties[13,14℄,ifthe
followingonditionissatised:
!>!
=1
1
; (4)
where is theorrelation lengthexponentof the
uni-formmodel. ! isthewanderingexponent
!= logj
2 j
log
1
; (5)
where
1 and
2
aretheeigenvaluesofthesubstitution
matrixdenedas
M=
1 b
b 1 0
(6)
Itiswellknownthat,forthesequenesdenedin(2.3),
orresponding utuations on the hierarhial latties
disussedabovewillbeirrelevantforb=2andrelevant
for b=3. So, ifwerestritourselvesto thesituations
b =p=2and 3 wean onvenientlyharaterize the
twodierentsituations, keeping onstant the valueof
thefrataldimensiond
f .
III Transfer matrix formulation
ThedetailsoftheTMmethodfortheevaluationof
ther-modynamifuntionsoffratalandhierarhiallatties
havebeenpresentedin earlierworks [5, 6℄, sothat we
will disuss here only its main steps. Sine the
hier-arhial lattiesare bounded by tworoot sites in any
generation G, the eet of allinterations onthe way
from one root siteto the other anbedesribed by a
2x2TMT
G
T
G
inorporatestheontributionoftheinteration
be-tweenallintermediatespins. Thisaneasilybedoneif
wereognizethat thematriesof twosubsequent
gen-erations, T
G and T
G+1
, have the same struture, so
thatamatriialmapexpressingT
G+1 =T G+1 (T G )an
beobtained. This implies,inturn, that thematrix
el-ements of T
G+1
an be expressed in terms of those of
T
G
by nonlinear maps. The derivation of suh maps
onstitutestheessentialstepfortheimplementationof
the method. It must be pointed out that any model
on a given lattie will obey the same matriial map,
whereas the map for the matrix elements will assume
dierent forms aordingto the harateristis of the
dierent models. The nonlinear maps an not be
in-tegrated exatly,but itisquiteeasytointegratethem
withanumerialgorithm.
The matrix elements are Boltzmann weights that
grow very quikly with G leading to numerial
over-ows. Tosidestepthis problem,it iswiseto usemore
onvenientvariables,thatareexpressedintermsofthe
matrix elements. Sowe dene, for any generation G,
the freeenergyperspinf
G
andorrelation length
G ,
in termsof
G and
G
,theeigenvaluesofT
G ,as f G = T NG log G ; G =M G =log(
G = G ): (7) whereN G
ountsthenumberoflattiesitesin
genera-tionG;and M
G =b
G
is theshortestdistanebetween
therootsites.
Themaps forthenewvariables, whih followfrom
the originalmaps forthematrixelements,arewritten
in termsof thesame set of variables in thepreeding
generation. In theiteration proess, the initial
ondi-tion isatemperaturedependent statewhih desribes
thesingleinterationbetweenthetworootspinsinthe
zero-th order (G =0) generation. The mapsare
iter-ated until the desired preision (usually 10 16
) is
ob-tained. Moreover, we an dene a generation
depen-dententropyandspeiheatbythederivativesofthe
spei free energy with respet to the temperature,
i.e., s
G
= f
G
=T and
G = T 2 f G =T 2 . Thus,
the enlargedset ofmaps inludes newomponentsfor
thederivativesoff
G and
G
,whih willdependonf
G
and
G
andonthesamederivativesin theformer
gen-eration. The same is valid for the derivatives of f
G
withrespettothemagnetield,sothatmapsforthe
magnetization and suseptibility anbederived. The
solutions of the maps are numerially very stable, as
evidenedbythereprodutionofknownpropertiesand
ritialexponentsofthehomogeneoussystems. The
ex-pliit formof themaps forthetwosituationsexplored
in thisworkisgiveninRef. [5℄.
IV Results
Theformalsolutionforthefreeenergyin RGindiates
f(t)=jtj 2 P lnjtj ln (8)
where t = jT T
j=T
and is the largest(thermal)
eigenvalueof the RGow lose to the ritial FP (or
otherritial set). Thefuntion P is anarbitrary
pe-riod 1 osillating funtion, whih inludes the
possi-bility P =onstant. To unoverand analyze the
log-periodi osillations, we integrate the maps in a very
narrow t-neighborhood of the ritial temperature T
, the value of whih an beevaluated within the TM
approahbyfollowingthevalueofT where!1. In
ordertofollowosillationsoverseveralordersof
magni-tude,wehoosedisretevaluesofT
n
suhthatt
n =t
n 1
isonstantandthevalueslog
10 (t
n
)areequallyspaed
onthelog
10
(t)axis. Fromtheresultingvalueswe
eval-uate thenumerialderivative oflog
10
(g) with respet
tolog
10
(t),wheregisanyofthethermodynami
fun-tions mentioned above, and graphit as a funtion of
log
10 (t).
We have analyzed models on the two
hierarhi-al latties disussed in Setion II, with b = p = 2
and 3 . As to the hoie of the sequenes ontrolling
theexhangeonstants,wehaveonsideredthe
follow-ingdistint situations: i)thehomogeneousmodels for
b=p=2and 3; ii) theperiod 2modeldesribedby
the sequene A ! AB, B ! AB on the lattie with
b = p = 2 ; iii) the aperiodi model (with irrelevant
utuations) indued by the sequene (3) with p =2
onthelattiewithb=p=2. Notethattheaseof
rel-evantutuations induedbythe samesequene with
p = 3 has already been treated on Ref [6℄. Typial
resultingplotsareshownin Figures1-4.
Figure1. Logarithmiderivativeofj (T) (T
)jwith
re-spet to log10(t) vs: log10(t) for the period 2 sequene
(A!AB;B !AB). Theverysmallvertial sale
nees-sarytoshowtheverysmallamplitudeofosillationsauses
makes it neessary to introdue a deay term in9. Here
andinallotherillustrations,openirlesindiatethe
Figure 2. Logarithmi derivative of with respet to
log10(t)vs:log10(t)forthehomogeneousmodelonthe
lat-tie withb=p=3:Here againaverysmallvertial sale
isneessaryinordertomaketheosillationsvisible.
Figure 3. Logarithmi derivative of with respet to
log10(t)vs:log10(t)forthehomogeneousmodelonthe
lat-tiewithb=p=2:Notethatithasthesameperiodasthe
urveinFigure1. Heretheamplitudeislargerthaninthe
formerases,sothatadeaytermisnotneessarytotthe
urve.
Toobtainquantitativevaluesfor theritial
expo-nentsandharaterizethemorphologyoftheosillatory
partof theurves,wetthedatawiththefuntion
y(x)=a+
x d
+ N
X
n=1 b
n
os(n!x+
n
); x=log
10 (t);
(9)
wherewehavetakenN =2inalmost alltting
proe-dures. TheresultsaresummarizedintheTable1. The
parametersanddin(9)havenopreisemeaning,but
this extra termis neessaryto obtain verypreise
re-sults for someof the funtions, expressed by thevery
low 2
values forall ttings. Theyare related to the
distanefromtheregionwherethepointsareevaluated
to theverysmallneighborhoodof T
wheretheurves
aredesribedonlybytheritialexponent(a)andthe
amplitudes and phases (b
n and
n
) of the osillatory
funtion. Forsomeurveswehavealreadyreahedthis
neighborhood,sothat0:
Figure 4. Logarithmi derivative of m with respet to
log
10
(t)vs:log
10
(t)fortheaperiodimodel(A!AB;B!
AA)onthelattiewithb=p=2:Notethattheperiodhas
doubled in omparison to that in Figures 1 and 3. Here
againtheamplitudeisverysmall.
IV.1 Periodi ases
Thepresentdisussion applies to situations i) and
ii). Thelearperiodiosillationsover3deadesshown
in Figures 1-3 have very small amplitudes (10 5
or
smaller) for the spei heat and orrelation length,
whileonlyforthemagnetisuseptibilitydoweobserve
largeramplitudes. Asthespeiheaturveis
hara-terizedbyauspatT
,theritialbehaviorisanalyzed
forthedierenej(T) (T
)j:Forlargervaluesoft,
theurveisstill farawayfromtheregionwhere ithas
theformofaonstantvaluesuperimposedonperiodi
osillations. Forsmallervaluesoft,round-oerrorsand
theverysmalldierenesbetweenneighboringfuntion
valuesinduesthepreseneoflargeutuationsin the
derivatives,so that theresultingpointsbeome
mean-ingless.
Theresults in Table 1orroboratethe known
val-ues for the ritial exponents (b = p = 2) as well as
those for (b=p=3) obtainedby Haddad[15℄ and in
our previous work [6℄. As visually antiipated by the
Figures,theomputedvaluesfortheosillation
ampli-tudeareindeedverysmall. Comparisonoftheseresults
with those obtainedfor relevant aperiodi osillations
indiates thatthelaterareatleast twoordersof
mag-nitude largerthanthe orrespondingvaluefor the
pe-riodiase. Theomparisonoftheontributionof the
twoFourieromponentsindiates that theosillations
b
2 =b
1 <10
2
. Theevaluation ofthe period of
osilla-tions indiate that ourresults arein best aordwith
those indiatedby RG analyses,where =1:678and
2:250 forb=p=2and3respetively.
ThegraphsandttingsfortheperiodiABase(ii)
indiate that it shares the main features of its
homo-geneous ounterpart. This is notsurprising assaling
argumentsindiatethat,afteranitenumberofsteps,
relatedtotheintrinsiperiodof thesequene,the
sys-tems evolves asymptotially to the same situation as
the homogeneousmodel. The aboveevaluation for an
expliitsituationorroboratestheexpetedresults.
IV.2 Aperiodi irrelevant ase
ThissituationisillustratedinFigure4forthe
mag-netization, but periodi osillations are found for all
thermodynami funtions. The values for the tting
parameters in Table 1indiate that the ritial
expo-nentsremainthesameasinthehomogeneousase. The
magnitude of theosillations has slightlyinreased in
omparisontotheformerase. However,themost
im-portantdiereneregardstotheperiodofosillations.
Weobservethatitsvaluehasdoubledinomparisonto
thepreviousvalue,i.e.,ifwewritethefreeenergyinthe
form(8),thenPbeomesaperiod2funtion. Tobring
P baktoaperiod1funtion,theeigenvalueinthe
de-nominator of (8) should be squared. In any ase the
physialmeaningisthesame: thesystemonlyreturns
to its original ongurationafter two renormalization
steps. Besidesthat,wealsoobservethat thenearly
si-nusoidalharater of theosillations hasbeenlost,as
now b
2 b
1
always. This onstitutes another
impor-tant hange in the morphology of the osillations,
al-readyobservedintheaseofrelevantutuationswith
T <T
. However,inthepresentase,itindiatesthat
some50%(ormore)oftheamplitudeoftheosillations
resultsfromthesystemtryingtofollowtheoriginal
interpretationfortheobservedperioddoubling: asthe
FP (and its dynami properties), whih depends only
on the lattie and the homogeneous model, is not
af-fetedby theaperiodi utuations,the ritial
expo-nentsremainthesame,buttheperiodofosillationsis
aetedbythewaytheJ
A andJ
B
ouplings
renormal-ize bakinto their own new images. This extra eet
seemstobenotinludedintheRGgeneralsolution(8).
Inanyase,theontraditionbetweenourresultsand
those predited by RG needs to be investigated more
deeply.
Thisbehaviorshould bealsoomparedto that
ob-served in the ase of relevant utuations, where the
FPloosesitsattrativemanifoldandtheperiod of
os-illationsbeomesrelatedtotheeigenvalueofaperiod
2yleintherenormalizationowdiagram. Inthis
sit-uation,asintheaseofaperiodisequene,ourresults
agreewiththoseofRG.Butitshouldbeobservedthat
inthisase,thenewperiod2ylediretlydependson
the utuations,whih onstitutesaruial dierene
totheformerase.
V Conlusions
Inthisworkwehavearefullyanalyzedtheemergene
of log-periodi osillations for periodi and aperiodi
(with irrelevant utuations) Ising models on
hierar-hial latties. Our investigation is based on the TM
method,whihleadstonumeriallyverypreisevalues
forthe thermodynamifuntions after theiterationof
a set of maps. Although these osillations have been
predited from the solution of the RG maps for
sys-temswithdisretesaleinvariane,theyhavenotbeen
expliited evaluated, exept fora few exampleswhere
relevantutuations arepresent. Indeed,spin models
onhierarhiallattieshavebeeninvestigatedbyalarge
numberof authors [12℄ but, to the best of our
knowl-edge,noexpliitevaluationsofosillations,asshownin
thepreviousSetion areavailableintheliterature.
Ourresultsshowthatthemagnitudeofthe
osilla-tionsisrathersmall,whihexplainsthefatthat they
havebeenoverlookedinthemanyinvestigationsofthe
model. Theperiod ofosillation is found to be in
a-ordane with that predited by RG, for the ase of
periodi(orhomogeneous)systems: itdependsonlyon
theeigenvalueoftheritialFP.Theosillationsare
al-mostsinusoidalforT bothbelowandaboveT
,as an
bemeasuredbyomparingthetwoFourieroeÆients
obtainedin thettingproedure.
The ase with aperiodi irrelevant utuations
presents manydierenes. Whereas the ritial
expo-nentsarestillontrolledbythesameFPasthe
homo-geneousase, ourinvestigationshowsaslight inrease
in themagnitudeoftheosillations,andadoublingof
the period. This result stands in opposition to that
oered by RG,aordingto whih theperiod remains
the same and is ontrolled only by the properties of
thehomogeneoussystem. Our resultsuggeststhat,in
theaseofirrelevantsequenes, theperioddependsas
wellonthemodulatingsequene. Thisissupportedby
thefatthattheseondFourieromponent,whihnow
osillateswith theperiodof the homogeneoussystem,
hasaontributionthatisofthesamemagnitudeasthe
rst omponent. We suggestthat theperiod doubling
may be aused by the fat that, whereas the
geomet-rial propertiesof the lattie andof the homogeneous
systemfavorsperiod1osillations,therenormalization
oftheJ
A andJ
B
onstantsbakto theiroriginal
on-gurationdemandsahighernumberofrenormalization
steps.
Aknowledgements- The author is muh indebted
to S.T.R. Pinho for helpfully disussions and
sugges-tions. TheworkwaspartiallysupportedbyCNPq.
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