thermodynamical fingerprints of fractal spectra
TRANSCRIPT
il
PHYSICAL REVIEW E OCTOBER 1998VOLUME 58, NUMBER 4
Thermodynamical fingerprints of fractal spectra
Raul O. Vallejos1,* and Celia Anteneodo2,†
1Instituto de Fı´sica, Universidade do Estado do Rio de Janeiro, R. Sa˜o Francisco Xavier, 524, CEP 20559-900, Rio de Janeiro, Braz2Instituto de Biofı´sica, Universidade Federal do Rio de Janeiro, Cidade Universita´ria, CCS Bloco G,
CEP 21941-900, Rio de Janeiro, Brazil~Received 25 March 1998!
We investigate the thermodynamics of model systems exhibiting two-scale fractal spectra. In particular, wepresent both analytical and numerical studies on the temperature dependence of the vibrational and electronicspecific heats. For phonons, and for bosons in general, we show that the average specific heat can be associatedto the average~power law! density of states. The corrections to this average behavior are log-periodic oscil-lations, which can be traced back to the self-similarity of the spectral staircase. In the electronic problem, evenif the thermodynamical quantities exhibit a strong dependence on the particle number, regularities arise whenspecial cases are considered. Applications to substitutional and hierarchical structures are discussed.@S1063-651X~98!13808-4#
PACS number~s!: 05.20.2y, 61.43.Hv, 65.40.1g, 61.44.Br
ysth
-rs
y
,
te
-
anau
adi
ears
lti-ie
ics
t
teac
traatncesificra-iledr a
s.e.,ificinof
dic-g-ell
thenicody-husford.ssll a
inctraac-ntiallingis
n
aed
I. INTRODUCTION
Quasiperiodic structures have been studied intensivelthe last 15 years. Aside from their purely theoretical interethese studies were in part motivated by the discovery ofquasicrystalline state@1# together with the possibility of experimental realization of quasiperiodic superlattices, fiachieved in 1985 by Merlin and collaborators@2#. In turn, therich properties of these structures encouraged the studsystems based on alternative sequences~e.g., Thue-Morseand hierarchical!, which, in spite of not being quasiperiodicstill exhibit deterministic~or ‘‘controlled’’ ! disorder, i.e.,they are neither random nor periodic. One of the most inesting features that many of these problems~either experi-mental or theoretical! display is a fractal spectrum of excitations. For instance, experiments on Thue-Morse@3# andFibonacci @2# superlattices have uncovered scale invarienergy spectra. Numerical analysis of linear chains of hmonic oscillators with hierarchical nearest-neighbor coplings and equal masses exhibit spectra related to the triCantor set@4# ~the Cantor-like structure is preserved evenmasses are also distributed in a hierarchical way@5#!. It hasalso been proved that the energy spectrum of a chain madidentical springs and of masses of two different kindsranged after the Thue-Morse sequence is a Cantor-like@6#.
Even though controlled disorder typically leads to mufractal spectra, in some cases only a few scales are sufficfor a satisfactory understanding of their thermodynamFor instance, the phonon spectrum of the chain in@4# isessentially a one-scale Cantor set; the electronic spectraarise from Fibonacci tight-binding Hamiltonians~either on-site or transfer! are governed by a couple of scale factors@7#.
Within this context, few-scale fractal spectra constitusimple prototypes for testing the thermodynamical implictions of deterministic disorder. As a first step in this dire
*Electronic address: [email protected]†Electronic address: [email protected]
PRE 581063-651X/98/58~4!/4134~7!/$15.00
int,e
t
of
r-
tr--ic
f
of-et
nt.
hat
--
tion, in Refs.@8,9#, one- and multiscale fractal energy specwere studied within Boltzmann statistics. It was shown ththe scale invariance of the spectrum has strong consequeon the thermodynamical quantities. In particular, the specheat oscillates log periodically as a function of the tempeture. Moreover, general scaling arguments and a detaanalysis of the integrated density of states allowed foquantitative prediction of the average value~which is relatedto the average density of states!, period and amplitude of theoscillations.
In this paper we extend the analysis of@8,9# to N-particlesystems described byquantumstatistics. We present resultconcerning the relevant thermodynamical quantities, ichemical potential, average particle number, and specheat. Our conclusion is that for phonons, and for bosonsgeneral, the Boltzmann scenario survives the inclusionquantum symmetries. That is, the crudest~power-law! ap-proximation to the density of states leads to a correct pretion of the average behavior of the specific heat. Loperiodic oscillations decorating the average value are wreproduced when the simplest nontrivial corrections topower-law density of states are considered. In the electrocase, the presence of a Fermi surface makes the thermnamical quantities very sensitive to the particle number, tpreventing the use of smooth approximations. However,special cases, interesting regularities can still be observe
The paper is organized as follows. In Sec. II we discusome general aspects of the problem. We make as weshort revision of previous results, which will be usefulsubsequent sections. In particular, we introduce the spethat will be considered and describe their significant charteristics. In Sec. III we treat the phonon case, whose essefeatures can be explained analytically by means of scaarguments similarly to the Boltzmann case. Section IVdevoted to Bose~general case! and Fermi statistics. SectioV contains the concluding remarks.
II. GENERAL CONSIDERATIONS
We consider a spectrum constituted by levels lying on(r 1 ,r 2) fractal set@10#. Such a spectrum can be construct
4134 © 1998 The American Physical Society
el
m
fhy
x-tw
dtio
ith
he
esro
.n
tral
-nnhpar-
at
lat-his
or-gths-at
za-uen-
ies
andsultscalg
al-c
n-
to
ra-c-
tresthe
fir
ls,
on
PRE 58 4135THERMODYNAMICAL FINGERPRINTS OF FRACTAL SPECTRA
recursively starting from an arbitrary discrete set of lev$e (0)% belonging to the interval@0,D#. The hierarchy$e (n11)% consists of two pieces that are obtained by copressing the set$e (n)% by factorsr 1 andr 2, respectively. Thelowest end of the first piece is positioned ate50, the upperend of the second piece is positioned atD ~see Fig. 1!. Thefractal that arises in the limitn→` has a~Hausdorff! dimen-siondf given implicitly by r 1
df1r 2df51. In the particular one-
scale case, i.e.,r 15r 2[r , one has the explicit resultdf52 ln 2/ln r. An alternative way of viewing the construction othe fractal is that in terms of gaps and ‘‘bands.’’ At hierarcn there are 2n21 gaps~in gray in Fig. 1!, which define 2n
‘‘bands’’ ~in white!. Whenn is increased by one, the preeistent gaps remain untouched, but each band splits intonew ones and a new gap appears.
We expect the thermodynamical properties associatethis spectrum to depend only on its hierarchical organizaand not on the specific pattern forn50 @9#. For this reason,and for the sake of notation simplicity, we will take$e (0)%5$0,D% ~so that there are 2n11 levels at generationn). Inaddition, we makeD51 and kB51 (kB stands for Boltz-mann’s constant! throughout the paper.
A very useful tool in describing Boltzmann systems wfractal spectra@8,9# is the integrated density of statesN(e)defined as
N~e
b
-ne
ultficlot--w
yticalan3%e
li-oeatoccocalirrec-tes
lyx-s
e.
y
d at
-
eitialed,
aler-us-useior.
a
-
4136 PRE 58RAUL O. VALLEJOS AND CELIA ANTENEODO
log periodicity of the spectral staircase, which is capturedthe smooth approximation~2!. In order to show that this isthe case let us first rewrite Eq.~6! as an integral:
c~T!51
2n11E0
1F e/2T
sinh~e/2T!G2
dN~e!. ~7!
Now we make two approximations in Eq.~7!. First, we re-place the exactN(e) by its smooth version~2!. Second, asour interest is the low-T regime, we extend the upper integration limit to infinity ~this is equivalent to considering aunbounded fractal spectrum!. After some simple algebra warrive at
c~T!'Tdd@a91b9cos~v ln T2f!1c9sin~v ln T2f!#.~8!
Here a95aa8, b95b(b82c8v/d), and c952b(c81b8v/d); the parametersa8, b8, c8 are given by
FIG. 2. Phonons in a two-scale fractal spectrum.~a! Log-logplot of the specific heat per atom vs temperature. The curvesparametrized by the scale factors (r 1 ,r 2) indicated in the figure.~b!c(T)/Td vs lnT. In all casesn59. Full lines correspond to numerical results~6! and dotted ones to our analytical prediction~8!,which is valid only for low temperatures~this is the reason why theanalytical curves do not saturate at high temperatures!.
y
F a8
b8
c8G5
1
4E0
`
dxxd11
sinh2~x/2!F 1
cos~w ln x!
sin~w ln x!G . ~9!
Equation~8! is the phonon analog of the Boltzmann res~4!. It displays explicitly the average behavior of the speciheat and the log-periodic corrections. In Fig. 2 we have pted the analytical curves~8! for the same set of scaling factors used for the numerically exact calculations. For lotemperatures, the agreement between the exact and analresults is very good@notice that the disagreement in the mevalue for the (0.30,0.15) case actually corresponds to aerror#. The scale factorr 1 is responsible for the period of thoscillations throughv522p/ ln r1 @as shown in Fig. 2~b!the period decreases asr 1 increases#. Moreover, asr 1 con-trols the gap scaling with respect toT50, smallerr 1 valuesimply comparatively bigger gaps, and, in turn, bigger amptudes. For fixedT, decreasingr 1 causes the spectrum tshrink towardse50, as a consequence the specific hgrows. These are the features observed by Petri and Ruin their paper on one-dimensional chains with hierarchicouplings @4#. We can now understand quantitatively theresults as being a direct consequence of log-periodic cortions to the pure power-law scaling of the density of sta~2!.
IV. BOSE AND FERMI STATISTICS
The statistical problem of particles with nonidenticalnull chemical potential stands at the next level of compleity, both from analytical and numerical points of views. Ain this case analytical derivations seem not feasible~exceptin limiting regimes!, so we resort to a numerical procedurFor a fixedaveragenumber of particlesN we extracted thechemical potentialm as a function of the temperature bsolving the equation
N5 (j 51
2n11
1
e~e j 2m!/T61, ~10!
where the sum runs over the levels of the fractal truncatehierarchical depthn. The sign 1/2 corresponds to theFermi/Bose case.~We are not taking into account spin degeneracies, which will not modify our results.! The numeri-cal solution of Eq.~10! is not a difficult task, however, somcare has to be taken with the choice of an appropriate invalue form. Once the chemical potential has been obtainthe specific heat can be calculated as theT derivative of theaveragetotal energyE(T),
E~T!5 (j 51
2n11
e j
e~e j 2m!/T61. ~11!
A. Bosons
Even though bosonic excitations with non-null chemicpotential do not seem to be of main relevance for the thmodynamics of superlattices, we decided to include a discsion on this topic for reasons of completeness, and becabosons in a fractal spectrum display a paradigmatic behav
re
-
min
al
f-
threoaearath
ino
mo-
-
the
en-
the
e-
k-is
ries
ns
tice.thef ace,
ar-
of
l-
las-
one
PRE 58 4137THERMODYNAMICAL FINGERPRINTS OF FRACTAL SPECTRA
The dependence of the fugacityz5exp(m/T) with tem-perature is illustrated in Fig. 3~a!. Several curves, parametrized by the average particle number~indicated in the fig-ure! have been drawn. First of all, Fig. 3~a! displays thefollowing limiting features. AsT decreases, and the systeevolves towards condensation, the chemical potentialcreases from negative values up to a limiting value closezero~we recall that our lowest energy ise50). The numberof particles in the condensate is given byN05z/(12z).Thus, for very lowT, together with largeN, one hasz'121/N. In the opposite limit of high temperatures and smN, one obtains from Eq.~10! z→N* [N/2n11. In Fig. 3~b!we have chosen a small value ofr 1 to show thatm indeedoscillates as the temperature gets through the scales ospectrum. In this sense, Fig. 3~b! can be thought as an amplified version of Fig. 3~a!.
The dependence of the specific heat per particle ontemperature is illustrated in Fig. 4. Shown are the exactsult and the numerical calculation, which uses the smoexpression~2! for the spectral staircase. For low tempertures the agreement is excellent. It is clear that the mvalue of the specific heat is indeed associated to the avelevel density and that it suffices to take into account only
FIG. 3. Bosons in a two-scale fractal spectrum. Fugacityz vsln T, for different values of particle numberN ~indicated in thefigure!. The chosen scales are (r 1 ,r 2)5(1/3,0.2) in ~a! and(0.02,0.2) in~b!. In both casesn57.
-to
l
the
e-
th-ngee
first nontrivial correction to the power-law scaling to explathe oscillations.@It is worth pointing out that this scenaribreaks down in the fermionic case~see later!#. We havetested that the curves in Fig. 4 do not change in the therdynamical limit of increasing bothN and the hierarchyn butkeepingN* 5N/2n11 fixed, except that the oscillatory behavior extends to lower temperatures of the order ofr 1
n ~thetotal number of oscillations of the specific heat is equal todepthn). The relative particle numberN* plays the role of adensity, given that the number of levels below a certainergy grows with the volume of the system. WhenN* issufficiently small and the temperature is high enough,curves tend to those for the Boltzmann statistics@8,9#. This isanalogous to the usual statement that the parameterl3r~wherel and r are the thermal length and the density, rspectively! tells how good is the classical approximation.
Although an analytical description of the problem is lacing, our numerical calculations show that log periodicityrobust enough to resist the inclusion of bosonic symmettogether with the restriction of particle conservation.
B. Fermions
The results presented in this section are valid for fermioin general, however, we assume that the fractal (r 1 ,r 2) cor-responds to the electronic spectrum of a certain superlatFrom the theoretical point of view, such spectra appear insimplest model for studying the electronic properties olattice, i.e., a stationary tight-binding equation, for instanin its transfer version:
t j 11c j 111t jc j 215ec j , ~12!
wherec j denotes the wave function at sitej and$t j% are thehopping matrix elements. If these hopping elements are
FIG. 4. Bosons in a two-scale fractal spectrum. Log-log plotthe exact specific heat per particlec(T) vs T, for different values ofaverage particle numberN ~full lines!. Dotted lines represent a caculation that uses the cumulative density of states~2!. The horizon-tal line is c(T)5d, which is the classical average value for lowT.At high temperature, the specific heat decays according to the csical 1/T2 law. Scale factors are (r 1 ,r 2)5(1/5,1/3). The fractal hasdepthn59. Base-5 logarithms were used to show that there isoscillation per ‘‘decade.’’
m
w-cc
dpre-
th
apInr-ls,
-ng
p
of
ul-
icalheave
dsginpar-l toap
at
f
to
e
eticles
for
4138 PRE 58RAUL O. VALLEJOS AND CELIA ANTENEODO
ranged according to the Fibonacci sequence, the spectruenergies$e% is essentially a fractal of the type (r 1 ,r 2) ~see,e.g., @7#!, independently of the boundary conditions. Hoever, the simple scaling is lost if the more general Fibonaclass@12# sequences are considered.
The numerical scheme used for computing the thermonamical quantities in the bosonic case can be easily adato the electronic problem. Figures 5 and 6 exhibit somelations among chemical potentialm, average number of particles N, and temperature for ther 15r 251/3 case. Finger-prints of the fractal spectrum are clearly seen independence ofm with temperature~Fig. 5!. In the limit ofzero temperature, for a given integer particle numberN, m
FIG. 5. Fermions in a fractal spectrum. Chemical potentialm asa function of log3 T for all integer particle numbers 1<N<2n11
5128 ~increasing from bottom to top!. We have only plotted thelower part of the full figure because it is symmetric with respectm50.5. Parameters arer 15r 251/3, n56.
FIG. 6. Fermions in a fractal spectrum. Log-log plot of thchemical potentialm vs the reduced number of particlesN* forfixed temperature. The curves correspond toT5329, 326, and 323
~smoother curves correspond to higher temperature!. Gray lines in-dicate the energy levels. We usedr 15r 251/3 andn54.
of
i-
y-ted-
e
takes the value corresponding to the middle of the g(eN ,eN11), as happens, e.g., in intrinsic semiconductors.order to keepN fixed as temperature grows, the Fermi suface moves in the direction of the lowest density of leveintegrated over an energy intervalT. The relative concentration of levels in the neighborhood of the gap is a fluctuatifunction of the scalekBT. This gives rise to an oscillatoryprocess that persists untilT overcomes the largest ga(log3 T'21). From this point on, ifN* .0.5 (N* ,0.5), mtends in a monotonic way to1` (2`). For smaller valuesof r 1, oscillations inm are magnified~not shown!, analo-gously to what happens with bosons.
Figure 6 displays the chemical potential as a functionthe reduced number of particlesN* for three values of fixedtemperature. This figure gives some insight into the difficties involved in solving Eq.~10!. For low ~fixed! tempera-tures,m(N) tends to the spectral staircase and the numerproblem gets relatively difficult. As temperature grows tstaircase gets smoother and numerics simplify. We htakenN to be continuous, but observe that for integerN andlow temperatures~compared to the gap size! the curves passthrough the middle of the gaps, as is clear in Fig. 5.
The specific heat in the fermionic problem depenstrongly on the position of the Fermi surface. Thus, we beby analyzing some simple cases where the number ofticles takes special values. For this purpose it is usefurecall the picture of bands and gaps of Fig. 1. Each gdetermines a special number of particlesN* , namely, therelative number of particles that would lie below that gapT50. For instance, the first-generation gap determinesN*51/2. The n52 gaps correspond withN* 51/4,3/4. Con-versely, the denominator and the numerator inN* indicaterespectively the generationn of a gap and the number ofilled bands ofnth generation, e.g.,N* 51/3251/25 corre-sponds to a gap thatappearedin the fifth generation and toone filled band of fifth generation.
FIG. 7. Fermions in a fractal spectrum. Log-log plot of thspecific heat versus temperature. The reduced numbers of parare N* 51/32,1/16,15/16,31/32, and the scale factorsr 151/3, r 2
51/9. The horizontal lines represent the classical average valuethe appropriate number of ‘‘hot’’ electrons (N* 51/32,1/16) orholes (N* 515/16,31/32).
ifiifs
heheeroiad
ro
t
atth
o
is
o
oth
ac7
th
omdan
be-ohth-d-,the.eat,or
anales
lesedfor
-lec-ro-nstes,the
b-ectnon-
ys-m-ex-thero-es.lesg-
tion
e
PRE 58 4139THERMODYNAMICAL FINGERPRINTS OF FRACTAL SPECTRA
A first example that illustrates the structure of the specheat is depicted in Fig. 7. There we consider a spectrum wtwo different scalesr 151/3, r 251/9, and the selected set onumbersN* 51/32,1/16,15/16,31/32. For the first two caseN* 51/32,1/16, the size of the gap is of the order of tFermi temperatureTF . Once the temperature overcomes twidth of that gap, the electrons can fully access the nband above, so that the mean level occupation drops appmately to 1/2. From then on the electrons behave essentas Boltzmann particles: the specific heat oscillates arounconstant average value, as expressed by Eq.~4!. The averagevalue and the period are governed by the scaling factor 151/3 through Eq.~3!. The number of oscillations is equal tthe generation indexn: four if N* 51/16 and five if N*51/32. The case of numbersN* 531/32,15/16 is equivalento having respectivelyN* 51/32,1/16holes and it is thencomplementary to the previous one. Remarkably, in the cof holes, the dominant scale isr 2. The reason for this is thar 2 rules the scaling with respect to the upper edge ofspectrum, and, in fact, with respect to the upper~lower! edgeof every band~gap!; in other words,r 2 is the relevant factorfor negative temperatures. Equations~3! and ~4! are stillvalid provided thatr 1 andr 2 are interchanged. According tthese arguments, the choicer 25r 1
2 implies that the period ofthe oscillations for holes is twice that for electrons, whichconfirmed by our calculations~see Fig. 7!. Horizontal linescorrespond to the Boltzmann average result, properly nmalized in the case of holes. Increasing the hierarchyn andkeepingN* fixed does not change Fig. 7 because the gaptop of the filled band determines the smallest scale offractal that can be resolved.
Other special particle numbers allow for observingmixed electron-hole behavior. Let us consider, for instanthe casesN* 57/64,57/64, corresponding, respectively, toand 57 filled bands of sixth generation~Fig. 8!. For N*57/64, the Fermi temperature is several scales biggerthe width of the seventh gap, so that the low-T part of thespecific heat is associated to hole excitations jumping frthe ~empty! eighth band down over the sixth-, fifth-, anfourth-generation gaps. These excitations are of Boltzm
FIG. 8. Analogous to Fig. 7 but for the particle numbersN*57/64,57/64~see text!.
cth
,
xtxi-llya
se
e
r-
ne
e,
an
n
nature and govern the specific heat until the temperaturecomes of the order ofTF . At this point electrons are able tjump up over the third-generation gap between the eigand ninth bands. The high-T (T.324) oscillations are associated to electronic excitations through the third-, seconand first-generation gaps. Similar considerations apply tocomplementary caseN* 557/64. The horizontal lines in Fig8 correspond to our predictions for the average specific hwhich take into account the effective number of electronsholes that contribute to the specific heat in each regime.~Thefact that the caseN* 57/64 presents one less oscillation thits complementary is due to an overlap of temperature scin the transition from hole to electron behavior.!
The general case of an arbitrary number of particshows a nontrivial mixture of the simple behaviors describbefore. However, the gross features of the specific heatarbitrary N can be qualitatively~and sometimes quantitatively! understood as being a reflection of sequences of etronic and hole excitations that alternate themselves in pducing the oscillatory patterns shown in Fig. 9. Formulatiobased on smooth approximations to the density of staanalogous to those made for bosons, are bound to fail infermionic case. The intrinsic discontinuity of the Fermi prolem, which is critically enhanced by the dual scaling respto the lower and upper edges of the gaps, is in essencecompatible with smooth approximations~with the exceptionof too special cases!.
V. CONCLUDING REMARKS
We have analyzed the quantum statistics of model stems exhibiting two-scale fractal spectra, with special ephasis on the structure of the specific heat. Our findingstend previous results on classical statistics to show thatthermodynamical manifestations of spectral fractality arebust with respect to the inclusion of quantum symmetriThe general scenario for Boltzmann and bosonic particcan be summarized as follows. In spite of the very framented structure of the real density of states, a formula
FIG. 9. Fermions in a fractal spectrum. Log-log plot of thspecific heat vs temperature for the numbersN525,77, and thescale factorsr 151/3, r 251/9 (n57).
acrgetrep
uin-nt
thalro
e-sy
sticby
ofntialm
e in-ect
Pq,isdee
4140 PRE 58RAUL O. VALLEJOS AND CELIA ANTENEODO
that starts from a smooth approximation but takes intocount the coarsest log-periodic fluctuations is sufficient fogood description of the specific heat and other averaquantities. In the case of fermions, even if a treatment thauniform in the particle number is not possible, many featuof the problem can be understood with the help of the simresults for the Boltzmann case.
Our analysis was limited to two-scale fractals. Previoresults@9# indicate that provided the scaling towards theferior limit of the spectrum is uniform, the inclusion of additional scales will not change our conclusions in what cocerns the bosonic case. Fermions, however, can feelscaling with respect to any point of the spectrum. So,complexity of the fermionic problem would be proportionto the number of relevant scales, in spite of uniform zeenergy scaling.
Let us finish by mentioning that the log periodicities dscribed in this paper may be observable in real physical
y
at-
A
lis
-adissle
s-
-hee
-
s-
tems, e.g., in the Fibonacci superlattices@2,18#. In fact, al-though our discussion was restricted to perfect determinisystems, both experiments and numerical simulationsToddet al. @18# indicate that the hierarchical organizationthe electronic bands may be preserved even if substaamounts of~random! disorder are added to the system. Froa more general point of view, Saleur and Sornette@19# havedemonstrated that the connection between discrete scalvariance and log-periodic oscillations is robust with respto the presence of disorder.
ACKNOWLEDGMENTS
We acknowledge the Brazilian agencies FAPERJ, CNand PRONEX for financial support. Also acknowledgedthe kind hospitality we received at the Centro BrasileiroPesquisas Fı´sicas, where part of this work was done. Wthank C. H. Lewenkopf for useful remarks.
e,
@1# D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, PhRev. Lett.53, 1951~1984!.
@2# R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, and P. K. Bhtacharya, Phys. Rev. Lett.55, 1768~1985!.
@3# Z. Cheng, R. Savit, and R. Merlin, Phys. Rev. B37, 4375~1988!.
@4# A. Petri and G. Ruocco, Phys. Rev. B51, 11 399~1995!.@5# J. C. Kimball and H. L. Frisch, J. Stat. Phys.89, 453 ~1997!.@6# F. Axel and J. Peyrie`re, J. Stat. Phys.57, 1013~1989!.@7# Q. Niu and F. Nori, Phys. Rev. Lett.57, 2057~1986!.@8# C. Tsallis, L. R. da Silva, R. S. Mendes, R. O. Vallejos, and
Mariz, Phys. Rev. E56, R4922~1997!.@9# R. O. Vallejos, R. S. Mendes, L. R. da Silva, and C. Tsal
e-print cond-mat/9803265.@10# See, e.g., T. Te´l, Z. Naturforsch. Teil A43A, 1154~1988!.
s.
.
,
@11# S. Alexander and R. Orbach, J. Phys.~France! Lett. 43, L625~1982!; R. Rammal and G. Toulouse,ibid. 44, L13 ~1983!; C.Tsallis and R. Maynard, Phys. Lett. A129, 118 ~1988!.
@12# X. Fu, Y. Liu, P. Zhou, and W. Sritrakool, Phys. Rev. B55,2882 ~1997!.
@13# J. M. Luck and D. Petritis, J. Stat. Phys.42, 289 ~1986!.@14# F. Nori and J. P. Rodriguez, Phys. Rev. B34, 2207~1986!.@15# J. P. Lu, T. Odagaki, and J. L. Birman, Phys. Rev. B33, 4809
~1986!.@16# Y. Liu and R. Riklund, Phys. Rev. B35, 6034~1987!.@17# C. Anteneodo and R. O. Vallejos~unpublished!.@18# J. Todd, R. Merlin, R. Clarke, K. M. Mohanty, and J. D. Ax
Phys. Rev. Lett.57, 1157~1986!.@19# H. Saleur and D. Sornette, J. Phys. I6, 327 ~1996!.