Critial Behavior of Ferromagneti Spin Models
with Aperiodi Exhange Interations
T. A. S. Haddad
, S. T. R. Pinho
, and S. R.Salinas
Institutode Fsia, UniversidadedeS~aoPaulo
CaixaPostal 66318,CEP05315-9790,S~aoPaulo,SP,Brazil
InstitutodeFsia,UniversidadeFederaldaBahia
40210-340,Salvador,BA,Brazil
Reeivedon15August,2000
We reviewreent investigations ofthe ritial behavior offerromagneti q-statePotts models on
a lass of hierarhial latties, with exhange interations aording to some deterministi but
aperiodisubstitutionrules. Theproblemisformulatedintermsofexatreursionrelations ona
suitableparameterspae. Theanalysisofthexedpointsoftheseequationsleadstoariterionto
gaugetherelevaneoftheaperiodigeometriutuations. Forirrelevantutuations,theritial
behavior remainsunhangedwith respet to the underlying uniform models. Inthe presene of
relevant utuations,anon-trivial symmetrixedpoint, assoiatedwiththe ritial behavior of
theuniformmodel,beomesfullyunstable,andthereappearsatwo-yleofthereursionrelations.
Asalinganalysis,supportedbydiretnumerialthermodynamialalulations,showstheexistene
ofanovelritialuniversalitylassassoiatedwithrelevantgeometriutuations.
Thedisoveryof quasirystals,and thedesignand
investigationofmagnetisuperlatties,providedstrong
motivation for the analysis of spin models with
ape-riodi exhange interations. Along the lines of the
Harris[1℄ riterion to gauge the inuene of quenhed
disorder on the ritial behavior of simple
ferromag-neti systems, Luk[2℄ has proposed that suÆiently
strong geometri utuations, assoiated with
deter-ministibutaperiodiinterations,mayalsohangethe
ferromagneti ritial behavior of the underlying
uni-formmodels. Luk'sheuristiriterionhasindeedbeen
hekedandonrmedin anumberofases(inluding
the quantum Ising hain[3℄, and the two-dimensional
Ising model[4℄).
In a series of reent publiations[5-10℄, we
onsid-ered ferromagneti Ising and Potts models, with
ape-riodi exhange interations aording to a variety of
two-letter substitution rules, on hierarhial
Migdal-Kadano latties[11℄. Taking advantageof the lattie
struture,wewereabletoderiveanexatexpressionfor
Luk'sriterion,andtoanalyzethenovelritial
behav-iorinthepreseneofrelevantgeometrialutuations.
Wenowpresentanoverviewofthese alulations.
As an example, onsider the period-doubling
two-lettersubstitution rule,
A!AB; (1)
B!AA; (2)
sothat the suessiveappliation ofthis ination rule
onletterAproduesthesequene
A!AB!ABAA!ABAAABAB!:::: (3)
Ateahstageofthisonstrution,thenumbersN
A and
N
B
, of lettersA and B, an beobtained from the
re-ursionrelations
N 0
A
N 0
B
=M
N
A
N
B
; (4)
withthesubstitutionmatrix
M=
1 2
1 0
: (5)
Theeigenvaluesofthismatrix,
1
=2,whihisthe
pe-riodof thetransformation, and
2
= 1,governmost
ofthegeometrialpropertiesofthesequene. Thetotal
numberofletters,atalargeorder nof these
onstru-tions, behaves asymptotially as n
1
. The geometri
utuationsareoftheorderj
2 j
n
. Itisthenonvenient
todenethewanderingexponentofthesequene,
!= lnj
2 j
ln
1
; (6)
that expressesthe asymptoti dependene of the
u-tuationswiththetotalnumberofletters,N N !
.
The period-doubling sequenean beused to
on-strut aperiodi spin models onhierarhial
To go beyond the simple Ising model, and introdue
anextraparametertoworkwith,weonsideraq-state
ferromagnetiPottsmodel, givenbytheHamiltonian
H= q X
(i;j) J
i;j Æ
i ;
j
; (7)
where
i
=1;2;:::;q,atallsitesofthehierarhial
lat-tie, J
i;j
> 0, and the sum refers to nearest-neighbor
pairs of sites (for q =2, we reoverthe Ising model).
The ouplings an take only two values, J
A and J
B ,
assoiated with a sequene of lettersprodued bythe
period-doubling substitution. In Fig. 1, we indiate
somestagesoftheonstrutionofthismodelona
sim-plediamondlattie(withb=
1
=2bondsandm=2
branhes). Notethatthehoieofthesameinterations
alongthebranhesofthediamondsisindeedmimiking
anaperiodilayeredstruture.
A
B
B
A
A
A
A
A A
A
A
A
A
B
A
A
B
B
A
A
B
Figure 1: Three suessive generations of the simple
dia-mondlattie (b =m =2). Letters A and B indiatethe
exhange parameters (whih are hosen aording to the
period-doublingsequene).
We may also onsider more general substitution
rules and basi "diamonds" with m branhes and b
bondsalongeahbranh,whihleadsto alattiewith
theintrinsiorfrataldimension
D=ln( mb)=ln(b): (8)
In eah one of these strutures, aperiodiity may be
implementedbyasubstitutionruleoftheform
( A;B)! A n1
B b n1
;A n2
B b n2
; (9)
with0n
1 ;n
2
b. Thesesequenesareharaterized
by asubstitution matrix with eigenvalues
1
= b and
2 =n
2 n
1
,whihleadsto thewanderingexponent
!= lnj
2 j
ln
1 =
lnjn
2 n
1 j
lnb
: (10)
Nowwedeimatetheinternaldegreesoffreedomof
thediamondstowriteexatreursionrelationsforthe
redued ouplings. For the q-state Potts model on a
lattie with b = 2 bonds per branh, and with
inter-ationsaordingtotheperiod-doublingrule,givenby
Eqs. (1)and(2),itiseasytowrite[8℄
x 0
A =
x
A x
B
+q 1
x
A +x
B
+q 2
m
; (11)
and
x 0
B =
x 2
A
+q 1
2x
A
+q 2
m
; (12)
wherex
A;B
=exp(qJ
A;B
) ,with =1=k
B T.
Inparameterspae,besidesthetrivialxedpoints
assoiatedwithzeroandinnitetemperatures,thereis
a non-trivial symmetri xed point, that governs the
ritial behaviorof theunderlying uniformmodel(see
theskethdrawninFig.2). Thelinearizationofthe
re-ursionrelationsaboutthissymmetrixedpointleads
tothematrixform
x 0
A
x 0
B
=C(q)M T
x
A
x
B
; (13)
whereC(q)isaq-dependentstruturefator,andM T
isthetransposeofthesubstitutionmatrix. The
eigen-valuesofthislinearform maybewritten as
1
=C(q)
1
=C(q)b; (14)
and
2
=C(q)
2
=C(q) !
1
=C( q)b !
; (15)
wherewehaveusedthedenitionof!,givenbyEq. (6).
Forirrelevantgeometriutuations,inwhihasethe
ritialbehaviorremainsunhangedwithrespettothe
underlying uniform model, it isrequired that
1 >1,
andj
2
j<1(seeFig. 2a). Therefore,weanwrite
1
=C(q)b=b y
T
; (16)
with the usual thermal ritial exponent assoiated
withtheuniformsystem,
y
T =
D
2
u
; (17)
where D, given byEq. (8),is thefrataldimensionof
the lattie, and
u
is the ritial exponent assoiated
with thedivergene ofthespeiheat oftheuniform
model[12, 13℄. From Eqs. (16)and (17),wehave the
struturefator
C(q)=b D
2
u 1
: (18)
Inserting this expression into Eq. (15)for the seond
eigenvalue,wehave
2 =b
D
2 u 1+!
; (19)
whihleadstoananalogofLuk'sriterionfora
hier-arhiallattie. Aswehavealreadymentioned,
utu-ationsareirrelevantforj
2
j<1. However,ifj
2 j>1,
whih orrespondsto afully unstablesymmetrixed
point,geometriutuationsarerelevant(seeFig. 2b).
Therefore, theritial behavior departs from the
uni-formaseifwehave
j
2
j>1 !!>1 D
2
whihisanexatriterionofrelevaneforthe
Migdal-Kadanohierarhiallatties. ForthePottsmodel on
thesimplediamondlattie,withb=2bondsandm=2
branhes,this riterionanbewritten as
q>q
=4+2 p
2=6:828427:::; (21)
where q
is thesameritialvaluefor therelevaneof
quenhed disorder aordingto a alulation by
Der-rida and Gardner[14℄ in the weak disorderlimit (also,
notethat
u
>0forauniformPottsmodelonasimple
diamondlattiewith q>q
).
Figure2: Shematirepresentationoftheparameterspae
andtheowdiagramsforirrelevant(a)andrelevant(b)
ge-ometriutuations. Thenon-trivialsymmetrixedpoint
of saddle-point haraterfor irrelevantutuations(a)
be-omesfullyunstableintherelevantase (b).
The reursion relationsare so simple that it turns
outtoberelativelyeasytoarryoutananalysisofthe
ritialbehaviorunderrelevantgeometriutuations.
ItisnotdiÆulttoshowthat,forq>q
,inwhihase
the non-trivialsymmetrixed pointisfullyunstable,
there is atwo-ylethat seemstobethe andidateto
dene anovellass ofritial behavior[8℄. Intermsof
the seonditerates ofthereursion relations,of whih
eahpointofthistwo-yleisaxedpoint,itdisplaysa
saddle-pointharater,withstable andunstable
mani-folds. Letususestandardsalingargumentstoanalyze
thenovelritialbehavior. Inthethermodynamilimit,
thereduedfreeenergyperbondanbewritteninthe
saling form[13,15℄
f(x)=g(x)+ 1
b 2D
f(x 00
); (22)
where g(x) is a regular funtion, x 00
is a seond
iter-ate of the reursion relations, b is the resaling fator
(numberofbondsofthediamondhierarhiallattie),
D is the frataldimension, and we usethe fator b 2D
beause weneed two iteratesto gobak tothe
neigh-borhood of theinitial pointin parameterspae. This
equationhastheasymptotisolution[15℄
f(x)jx x
j 2
P
lnjx x
j
ln
y
; (23)
wherex
isone of the pointsof the two-yle,
y is
thelargesteigenvalueofthelinearizationoftheseond
iterate ofthe reursionrelations aboutanyone of the
pointsoftheyle,P(z)isanarbitraryfuntionof
pe-riod1,andtheritialexponent,assoiatedwiththe
spei heat,isgivenby
=2 2 lnb
D
ln
y
=2 2 ln(mb)
ln
y
: (24)
All the alulations that we have performed, for a
q-statePottsmodel on avariety ofMigdal-Kadano
hi-erarhial latties, with dierent values of b and m,
andsuitablesubstitutionsequenes,pointoutthatthe
values of from this saling analysis are
unequivo-ally dierent from the values of
u
for the
orre-sponding underlying uniform models[8℄. For
exam-ple, for the period-doubling rule and the simple
dia-mond lattie (b = m = 2), with q = 7, the
two-yleis loated at(x
A ;x
B )
1
=(5:285:::;7:642:::), and
(x
A ;x
B )
2
=(6:697:::;4:750:::),with eigenvaluesofthe
seond iterate
1
= 3:933::: and
2
= 0:985:::, from
whih wehave = 0:0022:::, whih should be
om-paredwith
u
= 0:010:::. With q = 25, the
weaken-ing of the seond-ordertransition is even learer: the
two-yleisloatedat(x
A ;x
B )
1
=(6:942:::;234:34:::),
and (x
A ;x
B )
2
= (39:023:::;3:831:::), with eigenvalues
of the seond iterate
1
= 4:243:::and
2
=0:343:::,
fromwhihwehave=0:0817:::,tobeomparedwith
u
=0:404:::.
Totest the validity of the saling arguments, and
theroleofthetwo-yleastheresponsibleforthenew
ritial behavior, we have performed diret numerial
analysesofthesingularityofthefreeenergy[8℄. Inthese
problems,itiswellknownthatthereduedfreeenergy
an beexpressed as an inniteseries[13, 15℄. Forthe
Potts model on the b = 2 diamond-type lattie, with
theperiod-doublingrule,wehavetheseries
f(x
A ;x
B )=
1
X
1
(2m) n
1
3 ln
x ( n)
A +x
( n)
B
+q 2
+ 1
6 ln
2x (n)
A
+q 2
wherex ( n)
A;B
indiatethenthiterateofthereursion
re-lations. Asusual,weassumeuniformonvergene,and
dierentiate term by term to obtainthe spei heat
perbond. Theritialtemperatureanbedetermined
with high preision by making useof theexistene of
theparamagneti xed point at T =0,orresponding
tox
A;B
=1,whihausestheapparentdivergeneof
the seriesif summed withoutusing anyregularization
trik. Fixing the parameters q and m, and also the
strengths of J
A and J
B
, the ritialtemperaturethus
alulatedinfatloatesx
A;B
intheattrationbasinof
thetwo-yle. Aordingtotheexpetations, for
irrel-evantgeometriutuations,orfortheuniformmodel,
this method yields aritial temperature that loates
thesystemin thestable manifoldoftheuniform xed
point.
Thesingularityinthespeiheatanbeanalyzed
byttingthenumerialthermodynamidatatoa
fun-tionoftheformA+Bj(T T
)=T
j
overasomewhat
arbitrary saling region. For the irrelevant and
uni-formases,wehavealwaysobtainedverygoodttings,
in exellentagreementwiththevaluesof
u
predited
by thesaling theory aroundthe uniform xed point.
Of ourse, these tted values did not present any
de-tetable sensitivity onthe partiular valuesofJ
A and
J
B
. TheIsing model(q =2) onthesimpleb=m=2
diamondlattie, with exhange interations aording
to the period-doubling sequene, had been previously
and independently analyzedby Andrade[16℄, with the
sameonlusions. Forlargervaluesofq,inwhihases
the saling analysis predits positive values of
(al-though, of ourse, smallerthan the valuesof
u ), the
ttingspresentedexellentagreementwiththesaling
preditions. For weakersingularities (mainly negative
values of ), the tted values were always somewhat
bigger thanthesaling preditions,with better
agree-ment for inreasing values of q. For b = m = 2, and
q=7,weobtained= 0:005(4),tobeomparedwith
the saling result = 0:0022:::; for q = 25, we
ob-tained=0:08(2),tobeomparedwith=0:0817:::.
From the numerial alulations, wehaveseen an
os-illatorybehaviorof the spei heat asa funtion of
T. Theperiodofosillationisroughlygivenbythe
ar-gumentin Eq. (23),withbetteragreementfor
inreas-ing valuesof q. These log-periodiosillations, whih
arewellknownphenomenaassoiatedwithhierarhial
strutures,andindiatetheneedofamorerened
sal-inganalysis,havealsobeendetetedinthealulations
ofAndrade[16,17℄.
Inonlusion,wehaveshowntheexisteneofan
ex-atriteriontogaugetherelevaneofgeometri
utua-tionsinalassofspinsystemsonMigdal-Kadano
hier-behaviorremainsunhangedwithrespettothe
under-lyinguniform models. Inthepreseneofrelevant
u-tuations, thenon-trivial symmetrixed point, whih
is assoiated with theritial behaviorof theuniform
model,beomesfullyunstable,andthereappearsa
two-yle of thereursion relationsin parameterspae. A
standardsalinganalysis,supported bydiret
thermo-dynamial alulations,showsthe existeneof anovel
universalitylassassoiatedwiththerelevantgeometri
utuations.
ThisworkhasbeensupportedbytheBrazilian
agen-ies FAPESP,CAPES,andCNPq.
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