Study of the log-periodic oscillations of the specific heat of Cantor energy spectra

Transcription

1 Physica A 358 (2005) Study of the log-periodic oscillations of the specific heat of Cantor energy spectra Ana V. Coronado, Pedro Carpena Departamento de Física Aplicada II, E.T.S.I. de Telecomunicación, Universidad de Málaga, Málaga, Spain Received 14 February 2005 Available online 31 May 2005 Abstract Energy spectra with fractal structure are known to lead to a specific heat with log-periodic oscillations as a function of temperature. In this paper, we present a systematic study of the properties of these oscillations for both monoscale and multiscale energy spectra obtained from Cantor sets. We obtain how the amplitude of the oscillations depends on the structure of the spectrum. We also find that the amplitude of the oscillations above the specific heat mean value behaves differently to the amplitude of the oscillations below the mean value. The amplitudes of the latter oscillations has a limiting value given by a characteristic dimension of the spectrum. This asymmetry in the amplitudes produces strong non-harmonic behavior of the oscillatory regime when energy spectra with fractal structure characterized by small fractal dimension are considered. In addition, we also study the behavior of the specific heat when the energy spectrum has unity fractal dimension, i.e. corresponding to an energy spectrum without energy gaps, which are more similar to natural fractal energy spectra than usual Cantor sets. In this case we also find oscillatory behavior of the specific heat if certain conditions are satisfied. r 2005 Elsevier B.V. All rights reserved. Keywords: Log-periodic specific heat; Fractal energy spectrum Corresponding author. address: pcarpena@ctima.uma.es (P. Carpena) /$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi: /j.physa

2 300 A.V. Coronado, P. Carpena / Physica A 358 (2005) Introduction The study of the thermodynamic properties of physical systems with fractal energy spectra has been a topic of growing interest since the seminal work by Tsallis et al. [1], where the specific heat corresponding to an energy spectrum with the structure of the triadic Cantor set was studied. The most exotic property of the specific heat in this case is that it exhibits log-periodic oscillations around its mean value which is given by the fractal dimension of the energy spectrum. These results were generalized to more complex artificially generated fractal spectra [2,3] and to energy spectra corresponding to natural elements [5]. This oscillatory behavior has been also found in the specific heat of quasiperiodic systems, which are known to have a fractal energy spectrum [6], either modeled with simple 1D tight-binding hamiltonians [3,4] or as more realistic quasiperiodic superlattices [7]. Recent studies generalize the results to energy spectra derived from the logistic map [8], even when Fermi statistics is considered [9]. In all these works, oscillatory behavior of the specific heat is reported. However, there are two important properties of the oscillations of the specific heat in systems with fractal spectra that have not been systematically studied. First, it is usually assumed that these oscillations are harmonic (in log-scale). The reason for this belief is that this is the case, for example, for the triadic Cantor set and similar monoscale fractals. Nevertheless, this is not the case in general, and we show below that the oscillations in the specific heat can be strongly non-harmonic depending on the fractal properties of the energy spectrum. Second, there have been no studies of the amplitude of the oscillations as a function of the structure of the energy spectrum and we perform a systematic analysis of this dependence. In addition, we introduce fractal energy spectra based on Cantor sets with fractal dimension 1 and we study the behavior of the corresponding specific heat. In this case, the energy spectra do not have energy gaps, in contrast to standard Cantor set-based spectra. This kind of fractal energy spectra can be useful to model natural energy spectra with multifractal properties [5], which do not present energy gaps at all scales. The paper is organized as follows: in Section 2 we describe the method used to generate the energy spectra. In Section 3, we summarize briefly the main results obtained in the bibliography, to have a better perspective of our results. In Section 4 we study the oscillatory behavior of the specific heat of monoscale fractal spectra. In Section 5 we study similarly the multiscale case. In Section 6 we present the results obtained for Cantor energy spectrum with unity fractal dimension. Finally, we present our conclusions. 2. Generating the spectra We consider bounded spectra contained in the interval ½0; DŠ, and in the following, for the sake of simplicity, we fix D ¼ 1. We study two kind of fractal spectra, monoscale and multiscale, and in both cases we restrict ourselves to energy spectra formed by two spectral branches.

3 A.V. Coronado, P. Carpena / Physica A 358 (2005) Monoscale spectra: The generation process for the monoscale case (Fig. 1(a)) is the following: (1) divide the interval [0,1] into three segments, where the leftmost and rightmost segments have length p. (2) Eliminate the central segment. (3) Repeat the process in any of the two resulting segments, which now have a size p instead of a size unity. When this process is iterated n times, we say that we have a fractal spectrum in the nth step of generation (Fig. 1(a)). It is clear that in the nth step of generation, the spectrum consists of 2 n identical segments of length p n. The parameter p can take any value in the interval ð0; 1 2 Š, but the value p ¼ 1 2 leads to a non-fractal set in which all the energy levels of the spectrum are equally spaced. Since all the spectral branches are equal in size in any step of the generation process, we term this spectrum a monoscale spectrum. The fractal (box-counting) dimension for this set is simply d box ¼ ln 2 ln p. (1) Note that the classical Cantor set (or triadic Cantor set) corresponds to the case p ¼ 1 3, and it is the only homogeneous monoscale set, since for a generic p value different from 1 3, and although all the spectral branches are equal in size, the size of the energy gaps (the eliminated segments) is different to the size of the branches and therefore the resulting fractal is inhomogeneous. Multiscale spectra: The generation process of multiscale energy spectra is similar to the monoscale case, but now in the first step of the generation process the unity interval is divided in three branches where the leftmost and rightmost segments have lengths p 1 and p 2, respectively. After eliminating the central segment, the process is iterated in the left and right segments, (as we have shown in Fig. 1(b)) using the two different scale factors p 1 and p 2. The values p 1 and p 2 must satisfy the condition p 1 þ p 2 o1. As the generation step n is increased resulting in 2 n spectral branches, the diversity of sizes also increases leading to a multifractal set. In this case, the fractal (a) (b) P P n = 1 P 1 P 2 P 2 P 2 P 2 P 2 n = 2 P 1 2 P 1 P 2 P 2 P 1 P 2 2 P 3 P 3 P 3 P 3 P 3 P 3 P 3 P 3 n = 3 P 1 3 P 2 1 P 2 P 1 P 2 2 P 2 2 P 1 P P 2 1 P Fig. 1. (a) Monoscale spectrum with p ¼ 0:4. (b) Multiscale spectrum with p 1 ¼ 0:45 and p 2 ¼ 0:25. In both cases, we indicate the lengths of the segments for each step n of the generation process up to n ¼ 3.

4 302 A.V. Coronado, P. Carpena / Physica A 358 (2005) dimension d box of the set is obtained by solving [10] p d box 1 þ p d box 2 ¼ 1. (2) In this paper we consider only discrete energy spectrum, i.e. we take as the values of our energy spectrum the borders of the segments generated by the process described above for both the monoscale and multiscale cases. In general, due to the structure of the spectra, it is clear that in the nth step of the generation process, the spectrum will consist of N ¼ 2 nþ1 energy levels fe i g with the corresponding fractal structure. Once the energy spectrum is generated, the partition function Z n ðtþ can be evaluated Z n ðtþ ¼ XN i¼1 e be i, (3) where b ¼ 1=k B T, and for simplicity the Boltzmann s constant k B is set to unity in the rest of the paper. The internal energy U n ðtþ and the corresponding specific heat C n ðtþ can be obtained accordingly as 2 dðln ZðTÞÞ U n ðtþ ¼T ; C n ðtþ ¼ duðtþ dt dt, (4) where the index n indicates the generation step of the fractal spectrum. 3. Previous results The seminal work on this topic [1] studied the specific heat of an energy spectrum modeled as the triadic Cantor set (the monoscale case with p ¼ 1 3 ). The main results in Ref. [1] are that in the region of low T (corresponding to To1 in the units we consider), the specific heat presents log-periodic oscillations around the fractal dimension of the set (d box ¼ ln 2= ln 3), and with a logarithmic period given by ln 3, i.e. Cð3TÞ ¼CðTÞ in this T range. The oscillations are not only periodic (in logscale), but also harmonic. The number of oscillations found depends on the step of generation n of the fractal. When n increases, the oscillatory regime is extended toward smaller values of T. In the limit of large T, CðTÞ goes to 0 as T 2 due to the fact that the spectrum is bounded. For unbounded fractal spectrum, the oscillatory regime extends over the full range of T. Later works extended these results to other Cantor spectra [2,3]. In the general monoscale case, it is proved [3] that the logperiodic oscillations take place around the fractal dimension of the set. The period of the oscillations is given by the number of parts in which the unit interval is divided in the generation process. This is better illustrated with an example: let us generate a fractal spectrum by dividing the segment [0,1] into 5 segments of length 1 5, eliminate the second and fourth segment, and iterate the process. In this case, we have a fractal spectrum with three spectral branches. The corresponding specific heat presents logperiodic oscillations with a period given by 5, i.e. CðTÞ¼Cð5TÞ in the oscillatory regime, and the oscillations take place around the fractal dimension of the set,

5 A.V. Coronado, P. Carpena / Physica A 358 (2005) ln 3= ln 5. In general, we have that CðjTÞ ¼CðTÞ and d box ¼ ln m= ln j if the unit interval is divided into j parts and m parts are selected in the generation process. For the multiscale case, partly similar results have been found, but with two important differences. The first difference is that the log-periodic oscillations do not take place around the fractal dimension of the spectrum, but around the spectral dimension of the set, D, which is defined operationally from the behavior of the integrated density of states of the spectrum, NðEÞ. This function gives the number of energy levels below the energy E. It is usual to normalize NðEÞ, so that Nð1Þ ¼1, because E ¼ 1 is the highest energy of the spectrum. In Fig. 2 we show two examples of NðEÞ, corresponding to a monoscale and a multiscale fractal spectrum. Note that flat regions in NðEÞ indicate energy gaps, while steep regions in NðEÞ indicate the presence of many energy levels. In general, for fractal spectra, NðEÞ behaves approximately as a power-law: NðEÞ /E D (5) and the exponent D is precisely the spectral dimension. It has been shown that if NðEÞ behaves as in (5) then the average specific heat, hci equals D [5]. Actually, for fractal spectra NðEÞ behaves in double logarithmic scale as an oscillatory function around a power-law (see the inset of Fig. 2), and this is the reason why the oscillations of the specific heat takes place around the spectral dimension. The second difference is that the period of the log-periodic oscillations is given by the size of the first spectral branch: if this size is p 1 (as in Fig. 1(b)), then the period is given by 1=p 1, i.e. CðT=p 1 Þ¼CðTÞ in the oscillatory regime. In Section 5 we justify monoscale ( p = 0.28) multiscale ( p 1 = 0.21, p 2 = 0.37) 0.6 N (E) slopes: 0.444, Fig. 2. Integrated density of states NðEÞ for a monoscale and a multiscale fractal spectrum. Inset: the same in log log axis, where the straight lines represent fittings to power-laws. E

6 304 A.V. Coronado, P. Carpena / Physica A 358 (2005) these results with an analytic argument. To summarize, log-periodic oscillations have been found in the specific heat of fractal spectra, and the average value of the specific heat and the period of the oscillations have been studied. 4. Oscillations in monoscale fractal spectra The analytical derivation of the specific heat for a monoscale energy spectrum at the n step in the generation process is complicated, precisely because of the effects of the finiteness of the fractal set. Nevertheless, when an infinite fractal is considered, an analytical approach is possible. In this case, infinite fractal means that the fractal structure is propagated not only toward small scales in the interval [0,1] but also to the whole real energy axis. In Ref. [3] we have shown that if we divide the interval [0,1] into j parts, and we select the first and the last segment, and repeat this process to smaller (as usual) and larger scales (i.e., propagate to the whole energy axis), the specific heat for this unbounded fractal energy spectrum can be written as an infinite Fourier series. Although in that work we considered the case of j integer (and thus that any of the two spectral branches have a size of 1=j), in general the size of the branches can be any real number p with po 1 2. In this case, it is straightforward to note that the same Fourier series we introduced in Ref. [3] is also valid for non-integer j, simply substituting 1=j (the size of the branches) by p. In this way, we arrive at C 1 ðtþ ¼ ln 2 ln p 2 X 1 ½a k cosð2pkf ðp; TÞÞ b ln p k sinð2pkf ðp; TÞÞŠ, (6) k¼1 where for convenience, the following definitions have been used: Z 1 h 2pk ln h a k cosh 2 cos dh, (7) h ln p 0 Z 1 b k 0 h cosh 2 h 2pk ln h sin dh, (8) ln p f ðp; TÞ lnð2t=ð1 p 1ÞÞ. (9) ln p We use the name C 1 ðtþ in (6) to note that this is the expression of the specific heat when an infinite and unbounded fractal spectrum is considered. The analytical result (6) is used below to check our numerical results (see Fig. 4). Note that Eq. (6) contains the main properties of the specific heat in this case: the first term of the r.h.s. of Eq. (6) represents the average value of the specific heat, which is the fractal dimension of the set considered (see (1)). The second term represents the oscillatory part of the specific heat, and shows also the periodicity of the oscillations, which is given by 1=p, i.e., CðT=pÞ ¼CðTÞ. To show this, it is enough to show the periodicity of the first harmonic in the

7 A.V. Coronado, P. Carpena / Physica A 358 (2005) Fourier series: 0 cos 2pf p; T ln ¼ cos 2p p 2T=p p 1 ln ln p C A ¼ cos 2p 2T 1 p 1 ln p 11 ln p CC AA 0 0 ln ¼ cos 2p 2T 1 p 1 ln p! 1 1 C 2p C ¼ cosð2pf ðp; TÞÞ. A A ð10þ In previous works, the oscillations of the specific heat were considered to be harmonic. This is equivalent to assume that in Eq. (6) only the first term (k ¼ 1) of the Fourier series is relevant. That was the case in [1], where the triadic Cantor set (p ¼ 1 3 ) was studied, or in Ref. [3], where only fractal spectra with high values of p were studied. However, this is not true in general. This can be clearly seen by studying the behavior of the amplitude of the oscillations of the specific heat as a function of p. For convenience, we consider positive (A þ ) and negative (A ) amplitudes, defined as A þ C max ðtþ d box ; A d box C min ðtþ, (11) where C max and C min are the maximum and minimum specific heat values in each oscillation, and d box is the average value of the specific heat A + A 0.3 A + Amplitude A 1 Amplitude A d box p p Fig. 3. Behavior of A þ and A as a function of p. We include also (dashed line) the amplitude of the oscillation of the first harmonic (A 1 ). Inset: the same for low values of p, where the dashed line represents the fractal dimension d box of the corresponding fractal spectrum.

8 306 A.V. Coronado, P. Carpena / Physica A 358 (2005) In Fig. 3, we plot the behavior of A þ and A as a function of p. To obtain this figure, we have calculated numerically CðTÞ for 100 fractal spectra generated with 100 different p values uniformly distributed in the interval ð0; 1 2Š, each fractal spectrum generated with n ¼ 15 iteration steps. For each specific heat, we obtain numerically all the maxima and minima of the oscillations and average to obtain the average A þ and the average A. In this figure, we see how for large p, A þ and A are small and identical, because both curves overlap. Strictly speaking, A þ and A are 0 when p ¼ 1 2, i.e. when the spectrum is not fractal and d box ¼ 1. As p decreases, A þ and A increase together, up to certain value of p (around p ¼ 0:15) where they separate from each other. While A þ continues to increase as p decreases, A increases slower than A þ up to a maximum and then decreases as p tends to 0. The behavior of A þ and A indicates clearly that the oscillations are not harmonic for small p, because if that were the case both amplitudes would be identical in the whole range of p values. The reason for the behavior of A is clear: since the average value of the specific heat is d box, the value of A is limited precisely by d box. Note that A cannot be greater than d box because in this case the specific heat would be negative (see Eq. (11)). Actually, for large p this restriction does not influence A, because in this range A 5d box. But as p decreases and A increases, A approaches its limiting value d box,andasd box decreases as p decreases, A must decrease to avoid negative specific heat. This is shown in the inset of Fig. 3, where we also show the behavior of d box as a function of p. In this inset, note how for very small p values A! d box from below. This restriction does not apply to A þ, which can increase without producing negative specific heat. In Fig. 3 we also show in a dashed line the behavior of the amplitude of the first harmonic (A 1 ) in the Fourier series (6), which can be obtained straightforward from this series to give A 1 ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 1 ln p þ b2 1, (12) where a 1 and b 1 are evaluated numerically from (7) and (8). In the high p region, A þ, A and A 1 are indistinguishable, and therefore the log-periodic oscillations of the specific heat are essentially harmonic because the higher order Fourier coefficients a k and b k with k41 are negligible. The behavior of A 1 clearly indicates the region where the oscillations start to be not harmonic: when A þ and A departs from A 1, which is also the point where A þ and A depart from each other. This behavior is observed at p 0:15. For po0:15 the Fourier coefficients a k and b k are not negligible and higher harmonics are present in the oscillations. In Table 1 we show the values of the Fourier coefficients a k and b k up to k ¼ 3 of two fractal spectra, with p ¼ 1 3 and 0:01. In this latter case, the oscillations are strongly non-harmonic around the mean value of the specific heat. In Fig. 4, we show the specific heat for the cases p ¼ 1 3 and 0:01 obtained numerically by generating the corresponding energy spectra (up to n ¼ 13 and 5, respectively), as well as the corresponding Fourier series calculated from (6). In the case p ¼ 1 3, we have only used the first (k ¼ 1) coefficients of Table 1, and, as can be seen in the figure, the first term of the Fourier series is accurate enough to describe

9 A.V. Coronado, P. Carpena / Physica A 358 (2005) Table 1 Fourier coefficients of the specific heat of two fractal spectra with p ¼ 0:01 and 1 3 p a 1 b 1 a 2 b 2 a 3 b :154 0:0766 0:0177 0: : : p = 0.01 (up to k = 4) p = 0.01 (numeric) p = 1/3 (up to k = 1) p = 1/3 (numeric) 0.6 C n (T ) Fig. 4. Specific heat as a function of T for two monoscale spectra with p ¼ 0:01 and 1 3. We show the numeric results for finite step of the generation process and the analytical results obtained from Eq. (6). The horizontal lines represent in both cases the fractal dimension of the corresponding spectrum. T the oscillations, which are then essentially harmonic. For the case p ¼ 0:01, we show the result obtained using the coefficients up to k ¼ 4. Even considering the first four terms, still the oscillations are not exactly reproduced, indicating strong nonharmonicity. In all cases, the horizontal lines in Fig. 4 represent the average value (d box ) of the specific heats. Note that in the non-harmonic case p ¼ 0:01, the amplitude A þ above the average value is clearly larger than the amplitude A below it, in agreement with our results shown in Fig. 3 for this range of p. 5. Oscillations in multiscale fractal spectra When multiscale Cantor spectra are considered (as in Fig. 1(b)), instead of a single parameter controlling the fractal structure (p), this structure is given by two parameters, p 1 and p 2, which are the sizes of the two spectral branches in the first step of the generation process. In general, the specific heat of these spectra also presents log-periodic oscillations as a function of T (although not always, as we show below), but unfortunately there is not an analytical result equivalent to (6). The

10 308 A.V. Coronado, P. Carpena / Physica A 358 (2005) average value of the specific heat in this case is not given by the fractal dimension d box, but by the spectral dimension D defined in (5). For these spectra, D is given by D ¼ ln 2 ln p 1. (13) To show the validity of (13), and taking into account that D is defined in (5), it is enough to consider the structure of the integrated density of states NðEÞ at some special points. If we assume that NðEÞ is normalized (i.e., Nð1Þ ¼1), it is clear that Nðp 1 Þ¼ 1 2, Nðp2 1 Þ¼1=22, Nðp 3 1 Þ¼1=23 and so on. If we represent these points in a double log-scale plot, the slope of the linear fitting is precisely the one given in (13), and hence the average specific heat is this same value. As the spectrum is controlled by two parameters, p 1 and p 2, in order to study the behavior of the oscillations we proceed by fixing p 1 and varying of p 2, and vice versa. As in the monoscale case, we distinguish between positive and negative amplitudes, A þ and A. For each value of p 1 and p 2 we generate the corresponding fractal spectrum, calculate the specific heat and obtain the average A þ and A.InFig. 5, we show the behavior of A þ and A as a function of p 2 for four different values of p 1.In general, we can see how the amplitude of the oscillations increases as p 2 decreases, but its effect is not so critical because the amplitude of the oscillations increases approximately as a factor of 2 in the whole interval of variation of p 2 (p 2 2ð0; 1 p 1 Þ). As we can see in Fig. 5, for fixed and high values of p 1, the oscillations are symmetric around its mean value ða þ ¼ A Þ in the whole range of p 2. In the case of low p 1 values, we observe that A 4A þ for high p 2 values and A oa þ for low p 2 values, with an equality (A ¼ A þ ) appearing at an intermediate value of p P 1 = A + A Amplitude P 1 = 0.15 P 1 = 0.3 P 1 = P 2 Fig. 5. Behavior of A þ and A as a function of p 2 for four values of p 1. In all cases, p 2 varies in the range ð0; 1 p 1 Þ.

11 A.V. Coronado, P. Carpena / Physica A 358 (2005) A +, A, D p = p = p = D p 1 Fig. 6. Behavior of A þ and A as a function of p 1 for three fixed values of p 2. In all cases, p 1 varies in the range ð0; 1 p 2 Þ. Next, we fix p 2 and study the influence of p 1 on the amplitude of the oscillations. For each value of p 1, we generate the corresponding fractal spectrum up to n ¼ 15, obtain the specific heat and determine the averages of A þ and A. The results are shown in Fig. 6, where we show the behavior of A þ and A as a function of p 1 for three fixed values of p 2. As can be seen by comparing Fig. 6 with Fig. 3, the dependence of A þ and A on p 1 in the multiscale case is very similar to the dependence on p in the monoscale case. In general, for high values of p 1 both amplitudes are small and identical. As p 1 decreases, both amplitudes increases up to certain value of p 1 (which depends slightly on p 2 ) from where A þ and A depart from each other. From this point to smaller values of p 1, A þ increases monotonically while A reaches a maximum and decreases as p 1 tends to 0. As before, A is limited by the average value of the specific heat to prevent negative specific heat at some temperature. In this case, this limiting value is the spectral dimension D (13), which is also shown in a dashed line in Fig. 6, and for p 1! 0, A! D from below. We note also a slight dependence of the behavior of the amplitudes in the whole range of p 1 on the particular value of p 2, and in general the amplitudes are smaller for higher p 2. In general, p 1 can take any value in the interval ð0; 1 p 2 Þ. Although not shown in Fig. 6, the amplitudes of the oscillations become 0 (the oscillations disappear) when p 1 X0:5. Thus, if p 2 40:5, the specific heat of the corresponding fractal energy spectrum will always present oscillations for any value of p 1, because the maximum possible value of p 1 is 1 p 2 o0:5. In contrast, if p 2 o0:5, we can observe a transition from an oscillatory regime of the specific heat for any p 1 o0:5 to a non-oscillatory behavior when 0:5op 1 o1 p 2. The transition corresponds precisely to a fractal spectrum with p 1 ¼ 0:5, or equivalently, to D ¼ 1. Thus, fractal spectra with Do1 present log-periodic oscillations of the specific heat, while if D41 the oscillations disappear.

12 310 A.V. Coronado, P. Carpena / Physica A 358 (2005) Concerning the harmonicity of the oscillations, as there is no analytical result available, we have calculated numerically the Fourier transform of the oscillatory regime of the specific heat (when it exists, i.e. if p 1 o0:5) via the FFT algorithm. In general (not shown), we find that the oscillations are essentially harmonic when A ¼ A þ. Thus (see Fig. 5) for high values of p 1, we have harmonic oscillations while for low values of p 1, non-harmonic oscillations are observed, the transition being placed at the p 1 value for which the amplitudes A þ and A start to separate from each other. This transition from harmonic to non-harmonic oscillations depends slightly on p 2, as can be seen in Fig. 6. All these results show that the scale p 1 in multiscale fractal spectra has a very similar role to the scale p in the monoscale case: the period of the oscillations is given by 1=p 1 and 1=p, respectively, the average specific heat by D and d box, whose expressions are identical changing p by p 1, and the behavior of the amplitude of the oscillations are very similar (Figs. 3 and 6), except for the slight dependence on p 2. Even the absence of oscillatory regime takes place when d box ¼ 1(p¼ 1 2 ) and when D ¼ 1(p 1 ¼ 1 2 ). 6. Oscillations in energyspectra with unityfractal dimension In the previous sections, we have analyzed the oscillations of the specific heat of both monoscale and multiscale fractal energy spectra, as the ones shown in Figs. 1(a) and (b). In these spectra, we have that po 1 2 and that p 1 þ p 2 o1, respectively, or, in other words, the energy spectra present energy gaps which propagate at all scales in the generation process, leading to a self-similar structure. Actually, if p ¼ 1 2, we have that the fractal dimension is d box ¼ 1, and the spectrum consists of energy levels equally spaced in the energy axis, i.e., a non-fractal spectrum without gaps, for which there are no oscillations of the specific heat. In the multiscale case, when p 1 þ p 2 ¼ 1, using (2) we obtain that the corresponding fractal dimension is d box ¼ 1. As the fractal dimension is 1, this means that this type of spectrum has no energy gaps, or, equivalently, that the set generated in the n!1step of generation covers the whole ½0; 1Š energy interval. It is important to point out that many natural fractal spectra (for example, the energy spectrum of many elements of the periodic table) present multifractal properties [5,11], but in general the fractal dimension of these spectra is d box ¼ 1 because they do not present energy gaps at all scales. Thus, this kind of special Cantor sets for which p 1 þ p 2 ¼ 1 could be useful to model these natural spectra. Note that when Cantor sets with p 1 þ p 2 ¼ 1 are generated, the energy spectrum in the nth step of the generation process does not consist of 2 nþ1 energy levels, but of 2 n þ 1 levels, because the end of a spectral branch and the beginning of the next one corresponds to the same energy level, which is counted only once. To better visualize the structure of these spectra, in Fig. 7 we show the integrated density of states NðEÞ for four sets with p 1 þ p 2 ¼ 1 and four different values of p 1. In this case, there are no horizontal plateaus because there are no energy gaps, in contrast to standard Cantor sets (see Fig. 2). In the inset of Fig. 7, we present NðEÞ

13 A.V. Coronado, P. Carpena / Physica A 358 (2005) p 1 = 0.15 p 1 = 0.3 N (E) p 1 = 0.7 p 1 = E slopes: 0.365, 0.575, 1.94, 4.26 Fig. 7. Integrated density of states NðEÞ for four different energy spectra with d box ¼ 1. Inset: the same in log log scale. The straight lines represent fittings to power laws. 1.5 C n (T ) p = p = Fig. 8. Specific heat obtained for two fractal spectra with d box ¼ 1. We show the cases p 1 ¼ 0:15 and 0:6, both spectra obtained numerically up to n ¼ 10 in the generation process. The two horizontal lines represent the corresponding spectral dimension obtained from (13). T in log log scale to show that in all cases, NðEÞ is well described on average by a power-law with an exponent depending on the p 1 value. The question is if this type of energy spectrum without gaps presents oscillations in the specific heat. In Fig. 8 we show the specific heat for two energy spectrum with p 1 ¼ 0:15 and 0:6, below and above 0.5, respectively. We observe in general that if p 1 o0:5, the oscillatory regime is always present, while for p 1 40:5, the oscillations do

14 312 A.V. Coronado, P. Carpena / Physica A 358 (2005) not exist. When the oscillations exist, the period is given by 1=p 1 and the average value of the specific heat is again the spectral dimension D, given also by (13). This behavior is totally general, and coincides with the results we have found previously for multiscale Cantor energy spectra. Thus, we can conclude that the existence of gaps at all scales is not a restriction for the existence of log-periodic oscillations in the specific heat. 7. Conclusions We have studied the dependence of the log-periodic oscillations of the specific heat CðTÞ as a function of T of energy spectra described by fractal sets of Cantor type. We have shown that for monoscale spectra, an analytic expression for CðTÞ can be obtained, which shows that the average value of CðTÞ is the fractal dimension d box of the spectrum, and that the logarithmic period is given by p, the scale used to generate the fractal. Moreover, we also show that for low p values, the oscillatory regime becomes strongly non-harmonic because the amplitude of the oscillations below the average value is limited by the fractal dimension of the spectrum. In the multiscale case, similar results are obtained, but the role of p is now performed by p 1, the scale responsible for the low energy region of the fractal spectrum. In this way, the average value of CðTÞ is given by the spectral dimension (not d box ). The oscillations are strongly non-harmonic when p 1 is small, and the dependence of the amplitude of the oscillations on p 2 is not too important. The amplitude of the oscillations below the mean value of the specific heat is limited now by the spectral dimension D of the energy spectrum. The oscillatory regime disappears when p We have also studied special Cantor sets with d box ¼ 1, corresponding to energy spectrum without gaps, and that can be useful to model natural multifractal spectra. In spite of the absence of gaps, the behavior of this kind of spectra is identical to the standard multiscale case. References [1] C. Tsallis, L.R. da Silva, R.S. Mendes, R.O. Vallejos, A.M. Mariz, Phys. Rev. E 56 (1997) R4922. [2] R.O. Vallejos, et al., Phys. Rev. E 58 (1998) [3] P. Carpena, A.V. Coronado, P. Bernaola-Galván, Phys. Rev. E 61 (2000) [4] P.W. Mauriz, M.S. Vasconcelos, E.L. Albuquerque, Physica A 329 (2003) 101. [5] P. Carpena, A.V. Coronado, P. Bernaola-Galván, Physica A 287 (2000) 37. [6] P. Carpena, V. Gasparian, M. Ortuno, Phys. Rev. B 51 (1995) [7] E.L. Albuquerque, et al., Physica A 344 (2004) 366. [8] J.B. Soares Danyel, M.L. Lyra, L.R. da Silva, Phys. Lett. A 318 (2003) 452. [9] I.N. de Oliveira, M.L. Lyra, E.L. Albuquerque, Physica A 343 (2004) 424. [10] T.C. Halsey, et al., Phys. Rev. A 33 (1986) [11] A. Cummings, et al., J. Phys. B: At. Mol. Opt. 13 (2001) 2547.