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REVISTA MEXICANA DE F´ISICA S 54 (3) 48–55DICIEMBRE 2008

Scale invariance and criticality in nuclear spectra

E. Landa, I. Morales, C. Hern´ andez, J.C. L´ opez Vieyra, and A. Frank

Instituto de Ciencias Nucleares, Universidad Nacional Aut´ onoma de M´ exico,

Apartado Postal 70-543, M´ exico, D.F. 04510 M´ exico.

V. Vel´ azquez

Facultad de Ciencias, Universidad Nacional Aut´ onoma de M´ exico,

M´ exico, D.F. 04510 M´ exico.

Recibido el 15 de abril de 2008; aceptado el 9 de mayo de 2008

A Detrended Fluctuation Analysis (DFA) method is applied to investigate the scaling properties of the energy fluctuations in the spectrum

of48Ca obtained with (a) a large realistic shell model calculation (ANTOINE code) and (b) with a random shell model (TBRE) calculation.

We compare the scale invariant properties of the energy fluctuations with similar analyses applied to the RMT ensembles GOE and GDE. A

comparison with the related power spectra calculations is made. The possible consequences of these results are discussed.

Keywords: Quantum chaos; scale invariance; TBRE; DFA.

Se aplica el m´ etodo DFA (Detrended Fluctuation Analysis) para investigar las propiedades de escalamiento de las fluctuaciones de la energ´ ıa

en el espectro del48Ca obtenido con (a) un c´ alculo del modelo de capas realista (codigo ANTOINE) y con (b) un c´ alculo del modelo de

capas aleatorio (TBRE). Comparamos las propiedades invariantes de escala de las fluctuaciones de energ´ ıa con an´ alisis similares aplicados

a ensembles GOE y GDE de la teor´ ıa de matrices aleatorias (RMT). Se hace una comparaci´ on con c´ alculos relacionados de espectro de

potencias. Se discute las posibles consecuencias de esos resultados.

Descriptores: Caos cu´ antico; invariancia de escala; TBRE; DFA.

PACS: 05.45.Mt;24.60.Lz;52.25.Gj;74.40.+k;89.75.Da

1. Introduction

Our present knowledge of highly excited states in heavy nu-

clei is based on the connection with the eigenvalues of ran-

dom (chaotic) hamiltonians. On the scale of the mean level

spacing, the spectra of complex nuclei are statistically de-

scribed by Random Matrix Theory (RMT) [1]. This notion

was introduced by E. Wigner in the 1950s [2]. In particu-

lar, the probability distribution P(s) of the nearest-neighbor

spacing s agrees with the Wigner surmisei

P(s) = π/2se−π s2/4

of

conjecture [3] establishes that quantum systems whose clas-

sical analogs are chaotic, have a nearest-neighbor spacing

probability distribution given by RMT, whereas for sys-

tems whose classical counterparts are integrable, the nearest-

neighbor spacings are described by a Poisson distribution [4]

P(s) = e−s. Thus, a widely accepted criterion for a signa-

ture of quantum chaos is usually made in terms of the form

of P(s). Intermediate situations are analyzed by means of

interpolated distributions (see e.g. Refs 5 and 6).

Classical chaos, on the other hand, is a better understood

non-linear phenomenon, which gives rise to an unpredictable

time-evolution of the corresponding dynamical systems. In

particular, it is characterized by an intrinsic instability in the

orbits due to a high sensitivity to initial conditions. So, in-

stead of trying to make a precise prediction of individual tra-

jectories the aim of the theory of chaos is a description of

the space of possible trajectories and the evaluation of aver-

RMT.Furthermore,theBohigas-Giannoni-Schmit-

age quantities on this space. In general, the dynamical in-

stability of the orbits in a chaotic system is accompanied by

the occurrence of strange attractors with a fractal structure

in phase space (e.g. in the Lorenz model -see Fig. 1). The

origin of this fractal structure is related to the existence of a

rigid tree of periodic orbits (cycles) of increasing lengths and

self-similar structure [7]. The relation between the structure

of periodic orbits in phase space and RMT is established by

Gutzwiller trace formula [8]. Thus, at the quantum level we

would hope to find a signature of the fractality in the phase

space in the form of a scale invariance, or, in other words,

to identify the same kind of signature (or symmetry) in the

quantum regimeii

The notion of scale invariance appears in many different

phenomena. For example, in second order phase transitions,

it appears near the so called critical points where some phys-

ical quantities obey a power law behavior. In paricular, the

correlation lenghtiiiξ behaves like ξ ∼ |T − Tcrit|−ν, with

ν being the corresponding critical exponent. At the critical

temperature the correlation lenght ξ diverges and the system

has no characteristic scale, i.e. the system becomes scale in-

variant, and the correlation function behaves as Γ(r) ∼ r−p.

Power law behavior has been observed in the study of

chaotic time series, for example in the problem of a dripping

faucet [9], in heartbeat dynamics [10] and in many other phe-

nomena. Recently, it was found that the power spectrum of

the fluctuations of the eigenvalues of RMT ensembles and

nuclear shell (TBRE) model calculations exhibit a power law

behavior ∼ 1/f (with f being the frequency), whereas, for

the case of integrable systems it was found that the corre-

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SCALE INVARIANCE AND CRITICALITY IN NUCLEAR SPECTRA

49

sponding power spectrum behaves as ∼ 1/f2(see Ref. 11).

Thus, in the case of a system with a (parameter-dependent)

transition from a regular to a chaotic regime, like the hydro-

gen atom in an external magnetic field, we would expect to

have a power spectrum ∼ 1/f2associated with the energy

fluctuations at small magnetic fields, and ∼ 1/f for mag-

netic fields B ? 1 (in atomic units). To our knowledge, the

dependence of this transition on the external magnetic field

intensity has not been well understood so far. This problem

will be studied elsewhere.

The purpose of the present paper is to begin a study of the

self-similar (or fractal) properties of the energy fluctuations

in the spectrum of quantum chaotic systems. As a concrete

system we study the energy fluctuations in the spectrum of

48Ca, obtained with (a) shell model calculations with a re-

alistic interaction, and (b) with random shell model calcu-

lations (TBRE) both in the full fp shell. Large shell calcu-

lations are considered to exhibit the chaotic behavior found

in actual experimental spectra (see e.g. Ref. 23 and refer-

ences therein). We also carry out a comparison with the cor-

responding behavior of the energy fluctuations in the RMT

ensembles GOE and GDE. We shall also use a recently intro-

duced notion based on the analogy between energy fluctua-

tions of chaotic hamiltonians and chaotic time-series, and ap-

ply the method of detrended fluctuation analysis (DFA) [15],

which is designed to study the hidden fractal properties of

time series found in many natural phenomena.

2.Fractality and 1/f scale invariance

The concept of a fractal is associated with geometrical ob-

jects satisfying two criteria: self similarity and fractional

dimensionality. Self similarity means that an object is com-

posed of sub-units and sub-sub-units on multiple levels that

(statistically) resemble the structure of the whole object. A

FIGURE 1. Lorenz strange attractor having a fractal (Hausdorff)

dimension ∼ 2.06.

related property is scale invariance which can be thought of

as self-similarity on all scales. Thus, a fractal structure lacks

any characteristic length scale. This fractal structure is seen,

e.g., in the Lorenz attractor Fig. 1.

The 1/f behavior of the power spectrum found in

quantum fluctuations of the spectra of random hamiltoni-

ans [11,17] suggests that full quantum chaos can be asso-

ciated with a particular class of scale invariance. Namely, a

scale invariance for which the auto-correlation function be-

comes (approximately) scale independent. Such situation oc-

curs for a power spectrum with a power-law (scale invariant)

behavior ∼ 1/fβat the critical value β = 1. A demon-

stration in the continuum case is the following: suppose

that the power spectrumivof a given time series has a 1/f

behavior, i.e.

S(f) = 1/f .

(1)

Since the Fourier Transform of the power spectrum is iden-

tical to the autocorrelation function C(τ) (Wiener-Khinchin

Theoremv) we have:

C(τ) = F−1(S(f)) = F−1(1/f).

Now, if we make an arbitrary scale transformation in the time

domain (i.e. τ → aτ, a ∈ R+) we have

C(aτ) = F−1

(2)

?1

a(S(f/a)

?

= F−1

?1

a×a

f

?

.

(3)

Thus

C(aτ) = C(τ).

(4)

Here, we have used the scaling property of Fourier Trans-

forms, which is strictly valid only in the continuum case. For

discrete time series, there are other tools for studying their

scale invariant properties, including the DFA method [15]

(see below). In fact, 1/f behavior (referred to as flicker or

1/f noise) occurs in many physical, biological and economic

systems, meteorological data series, the electromagnetic ra-

diation output of some astronomical bodies, and in almost

all electronic devices. In biological systems, it is present in

heart beat rhythms and the statistics of DNA sequences. In

financial systems it is often referred to as a long memory ef-

fect. There are even claims that almost all musical melodies,

when each successive note is plotted on a scale of pitches,

will tend towards a 1/f noise spectrum.

3. Spectrum fluctuations as time-series

The fluctuations in a quantum spectrum are obtained by an

unfolding procedure, i.e., by substracting the gross features

of the spectrum which can be modeled by a smooth function.

In essence, this procedure consists in mapping the spectrumvi

Eiintoadimensionlessspectrum?i, havingameanlevelden-

sity of 1:

Ei → ?i≡¯ N(Ei),

(i = 1,...N).

(5)

Rev. Mex. F´ ıs. S 54 (3) (2008) 48–55

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E. LANDA, I. MORALES, C. HERN´ANDEZ, V. VEL´AZQUEZ, J.C. L´OPEZ VIEYRA, AND A. FRANK

where¯ N(Ei) is a smooth function fitviiof the staircase-like

cumulative density function N(Ei) (see e.g. Ref. 8). In par-

ticular, the nearest neighbor spacing (NNS) is calculated as

si= ?i+1− ?i,i = 1,...N − 1, and ?s? = 1. The spectrum

fluctuations can be defined by the quantity

δn=

n

?

i=1

(si− ?s?) = [?n+1− ?1] − n?s?.

(6)

The stochastic discrete function δnmeasures the deviations

of the distance between the first and the (n + 1)-th unfolded

states, with respect to the corresponding distance in a uni-

form (equally spaced) sequence having a unit level distance

?s? = 1. The sequence (6) can be formally interpreted as a

discrete “time series” (see e.g. Ref. 11). In order to under-

stand the scaling properties of the fluctuations (6), we use the

detrended fluctuation analysis (see below).

A standard measure for the deviation from equal spacing

is the Dyson-Metha rigidity function [16]

∆3(L;α) =1

LMinA,B

α+L

?

α

?N(E) − AE − B?2dE , (7)

where A,B give the best local fit to N(E) in the observa-

tion window α ≤ E ≤ α + L. The harmonic oscilla-

tor corresponds to the minimum value ∆3 = 1/12 (max-

imum rigidity), while a completely random (uncorrelated)

spectrum with a Poisson distribution has an average (over α)

∆3(L) = L/15 (see e.g. Ref. 8). The case of a GOE

spectrum with a Wigner-like NNS probability distribution is

an intermediate case and the rigidity function has the form

∆3(L) = 1/π2(logL−0.0687). Ithas beenshowninRef. 17

that the rigidity function (7) is related to the DFA method.

In particular, Santhanam et al. [17] have applied the DFA

method to RMT ensembles as well as to the spectra of heavy

atoms.

4.Detrended fluctuation analysis (DFA)

DFA is a method which allows the investigation of long range

correlations and scaling properties in a random time series. It

was first introduced in studies of DNA chains [15]. In the fol-

lowing we make a brief description of the DFA method (for

more details we refer the reader to the original paper [15]).

A time series δ(t) is self similar if the statistical proper-

ties of the full time series and the statistical properties of any

rescaled subinterval of it, satisfy the scaling relation

δ(t)

PDF

= aαδ

?t

a

?

,

(8)

where a is the scale factor in the time axis (aαis the corre-

sponding vertical scaling factor). The exponent α in Ref. 8 is

defined as the self-similarity parameter. We emphasize that

the equality in Ref. 8 is understood as indicating the same

probability distributions (PDF).

Let δ(i), i = 1...N be a time series. The DFA analysis

of δ(i) begins by defining an integrated time series

ψ(n) =

n

?

i=1

[δ(i) − ?δ?],

(9)

with ?δ? being the average (expectation value) of δ. Then the

integrated time series is divided into boxes of equal length ?,

where a linearviiileast-squares fit ψ?(n) (trend) is made. The

difference (r.m.s.) between the integrated time series and the

fit is measured by the detrended fluctuation

?

N

n=1

F(?) ≡

?

?

?1

N

?

[ψ(n) − ψ?(n)]2.

(10)

This fluctuation can be calculated for all scale factors (or box

sizes). In a log-log plot, a linear relationship between the

fluctuation and the box size will indicate a scaling (power

law) behavior. In this case the slope αDFAin the log[F(?)]

vs log[?] plot can be used to characterize the scaling prop-

erties (8) of the original time series since α = αDFA in

Ref. 8. As an example, if there is no correlation among

the points in the original time series δ(i), i.e. the autocor-

relation functionixC(τ) ≡ 0, for any time-lag τ ?= 0, the

time series behaves as white noise and the integrated time

series ψ(n) corresponds to a random walk characterized by

αDFA= 0.5 (see [18]). Time series with short range (expo-

nentially decaying) correlations C(τ) ∼ e−τ/τ0, τ0being the

characteristic scale, are also characterized by αDFA ? 0.5

although some deviations from αDFA? 0.5 may occur for

small window sizes. Of special interest are the so called per-

sistent (long time memory) time series for which the auto-

correlation function has a power-law behavior C(τ) ∼ τ−γ.

They are characterized by values 0.5 < αDFA < 1.0, the

relationship between γ and αDFAbeing γ = 2 − 2αDFA.

The power spectrum of the corresponding time-series also

displays a power-law (scale invariant) behavior S(f) ∼ 1/fβ

with β = 1 − γ = 2αDFA− 1. In particular, for time series

with 1/f-noise (β = 1) αDFA= 1 (see e.g. Ref. 19).

5.Results

We have applied the DFA method and performed a spectral

analysis to the energy fluctuations in the spectrum of48Ca.

For comparison purposes we have also applied the analysis

to the case of RMT ensembles GOE and GDE. In all cases

the unfolding (5) to the spectrum was done (for simplicity)

with a polynomial fit. After a careful analysis, a degree-7

polynomial fit was used in each casexHowever, the unfold-

ing is a delicate procedure when defining the energy fluc-

tuations [11]. It can lead to wrong conclusions when not

properly done. In particular, the results are rather sensitive

to the degree of the polynomial in a polynomial fit to¯ N(Ei)

Eq. (5). This fact has been discussed in TBRE calculations in

Ref. 13. In our analysis we have suitably removed the tails of

Rev. Mex. F´ ıs. S 54 (3) (2008) 48–55

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SCALE INVARIANCE AND CRITICALITY IN NUCLEAR SPECTRA

51

the spectrum to avoid a strong dependencies of the results in

the polynomial fit.

In the present calculations we have used the dfa C-code

(translation of Peng’s original fortran code [20]) with a linear

detrending option. The minimal box size used was 4, and the

maximal box size was N/4, with N the number of points in

the time series. The results of the analysis are presented in

the following paragraphs.

5.1.Realistic shell model calculations

Large shell model calculations with realistic interactions

(KB3) [21] were performed in the full fp shell for48Ca in the

subspaces Jπ= 0+,1+,...8+by means of the ANTOINE

code [24]. Within each subspace we calculated the energy

fluctuations following the definition (6) and applied a linear

DFA analysis. The value of the self-similarity parameter α

are found to be very close to 1, the largest deviations being

∼ 10%). The energy fluctuations represented by the time

series δnand its integrated form ψn, are shown in Figs. 2a

and 3a, respectively, for the case of the subspace Jπ= 0+.

The behavior shown in these figures is typical of all cases

studied with shell model calculations with realistic interac-

tions.

Figure 4a shows the results of the DFA analysis for the

case of the Jπ= 0+subspace. This case is particularly in-

teresting since this subspace contains only 347 energy levels.

It is quite remarkable that, even in this case, the trend of the

fluctuations is well approximated by a linear scaling in the

log-log plot in the whole domain of window sizes giving a

self-similarity parameter α = 0.97. Larger calculations show

even better linear scalings. The results of the DFA analysis

are summarized in Table I.

An α parameter close to 1 indicates an almost per-

fect non-trivial scale invariance.

β = 2αDFA− 1, we conclude that the power spectrum ex-

hibits a very approximate 1/f behavior. This is confirmed

Using the relation

TABLE I. Self similarity parameter α obtained using a linear DFA

method and the β exponent in the power spectrum of the energy

fluctuations in the shell model calculations of48Ca with realistic

interactions in different subspaces Jπ. The dimension N of each

subspace is also shown.

48Ca

Jπ

0+

1+

2+

3+

4+

5+

6+

7+

8+

αβN

347

880

1390

1627

1755

1617

1426

1095

808

0.969

0.998

1.013

1.020

0.985

0.916

1.077

1.095

0.964

1.008

1.090

1.046

1.183

1.127

1.198

1.137

1.180

1.031

FIGURE 2. Time series δn of the energy fluctuations in a shell

model calculation of the spectrum of48Ca (Jπ= 0+states) with

(a) realistic interactions, (b) with random interactions (TBRE), and

with RMT ensembles (c) GOE and (d) GDE. For the later the same

dimension as for the Jπ= 0+subspace was used. The time (hor-

izontal) axis represents the index of the ordered unfolded (dimen-

sionless) energy ?nand the vertical axis represents the correspond-

ing energy fluctuation δn i.e. the difference of the n-th unfolded

energy ?nwith respect to the n-th energy level in an equally spaced

spectrum with unit energy distance. Notice that the scale for the

fluctuations in GDE is about 4 times larger than for the shell model

calculations.

by the corresponding power spectrum calculations, shown in

Fig. 5a, where we find an exponent β = 1.008. The power

spectrum depicted in Fig. 5a shows the typical behavior in

all shell model calculations with realistic interactions: there

is a rather large spread in the Fourier amplitudes from a linear

scaling.

Rev. Mex. F´ ıs. S 54 (3) (2008) 48–55

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E. LANDA, I. MORALES, C. HERN´ANDEZ, V. VEL´AZQUEZ, J.C. L´OPEZ VIEYRA, AND A. FRANK

FIGURE 3. Integrated time series ψn(Eq. (9)) for the energy fluc-

tuations in shell model calculation of the spectrum corresponding

to48Ca (Jπ= 0+states) with (a) a realistic interaction, (b) with

random interactions (TBRE), and with RMT ensembles (c) GOE

and (d) GDE. For the cases of GOE and GDE the same dimension

as for the Jπ= 0+subspace was used. The time (horizontal) axis

represents the index of the ordered unfolded (dimensionless) en-

ergy ?n and the vertical one the corresponding integrated energy

fluctuation ψn. Notice that the scale for the fluctuations in GDE is

about 6 times larger than for the shell model calculations.

The observed spreading is seen independently of the size

of the spectrum subspace. We find in all cases that the DFA

method is a more robust procedure than the direct calculation

of spectral power when analyzing actual experimental data.

5.2.TBRE shell calculations

In the present study we have also applied the DFA method to

the energy fluctuations of the Two Body Random Ensemble

(TBRE) [22] shell model calculations for48Ca in the sub-

space Jπ= 0+. For this calculations we have used 25 sets of

energy levels.

FIGURE 4. Integrated time series ψn(Eq. (9)) for the energy fluc-

tuations in shell model calculation corresponding to the spectrum

of48Ca (Jπ= 0+states) with (a) a realistic interaction, (b) with

random interactions (TBRE), and with RMT ensembles (c) GOE

and (d) GDE. For the later the same dimension as for the Jπ= 0+

subspace was used. The time (horizontal) axis represents the index

of the ordered unfolded (dimensionless) energy ?nand the vertical

one the corresponding integrated energy fluctuation ψn.

The energy fluctuations represented by the time series δn

and its integrated form ψnare shown in Figs. 2b and 3b, re-

spectively. The self similarity parameter was calculated by an

averagingprocedureovertheDFAresults, anditwasfoundto

be α = 1.01. This value is very similar to the value of the self

similarity parameter obtained in the case of realistic calcula-

tions (see Table I). Figure 4b shows the averaged results of

the DFA analysis. The linear behavior of these results is very

striking. Only for very large window sizes (n ? N/4 ? 87)

we can see a slight deviation from linearity. Since the present

analysis was done for the case Jπ= 0+which has the small-

est dimensionality in the fp shell model calculations, it is

Rev. Mex. F´ ıs. S 54 (3) (2008) 48–55

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SCALE INVARIANCE AND CRITICALITY IN NUCLEAR SPECTRA

53

FIGURE 5. Linear fit of the Power Spectrum (log|Fk|2vs logk) of

the energy fluctuations in48Ca (Jπ= 0+subspace) obtained with

(a) a realistic interaction, (b) with angular momentum-preserving

random interactions (TBRE), and with RMT ensembles (c) GOE

and (d) GDE. Here |Fk| is the Fourier amplitude corresponding to

the frequency k in the discrete Fourier transform of the time series.

natural to expect a similar behavior as in Fig. 4b for larger

subspaces.

In Ref 11 Rela˜ no et al. performed the first study of the

behavior of the power spectrum of the energy fluctuations of

TBRE random shell model calculations for24Mg and32Na

and found that they obey a 1/f scaling. Our own power spec-

trum calculations of the energy fluctuations in the subspace

Jπ= 0+of the spectrum of48Ca Fig. 5b shows a behavior

similar to the one obtained by Rela˜ no et al. Ref. 11. En-

ergy fluctuations in TBRE calculations are characterized by

a reduction of the spreading of the Fourier amplitudes in the

TABLE II. Comparison between shell model calculations with (a)

realistic interactions, (b) random interactions (TBRE) and (c) GOE

calculations in the spectrum subspace Jπ= 0+of48Ca.

αβ

48Ca,Jπ= 0+

(Shell Model)

(TBRE)

N = 347

N = 1000

N = 347

N = 1000

0.969

1.003

0.951

0.942

1.338

1.398

1.008

0.987

0.998

1.069

1.604

1.786

GOE

GDE

power spectrum. This is an advantage of the averaging pro-

cedure. The power spectrum fit is shown in Fig. 5b where it

can be seen that in this case the linear fit in the log-log plot

adequately describes the 1/fβbehavior of the power spec-

trum. A different situation was observed in the case of the

corresponding shell model calculations with realistic interac-

tions depicted in Fig 5a. The value obtained for the scaling

exponent was β = 0.99 (see Table II) which implies a scale

invariance of the energy fluctuations in the TBRE calcula-

tions.

5.3. GOE

For comparison purposes we applied the DFA method to the

energy fluctuations in the case of GOE. In order to make a

fair comparison we considered a GOE with the same dimen-

sion as the case of the subspace Jπ= 0+, where both types

of shell model calculations (with realistic and random inter-

actions) were used. A set of 25 matrices in the ensemble was

used. The energy fluctuations represented by the time series

δnand its integrated form ψnare shown in Fig. 2c and Fig. 3c

respectively. With the results of the DFA analysis we ob-

tained a value for the self similarity parameter α = 0.95 (see

Table II), consistent with α = 1. This 5% deviation from the

expected value α = 1 is probably due to the unfolding pro-

cedure used in the analysis. This is also suggested by the fact

that in larger GOE calculations with a dimension N = 1000

a similar deviation from the value α = 1 is observed. It is

important to recall that in the limit N → ∞, the cumulative

function N(E) follows a semicircular law. However, even

for the case N = 1000 we observe significant deviations.

For the time being appropriate unfolding will be discussed

elsewhere [14].

The corresponding power spectrum calculations, on the

other hand, give a scaling exponent β = 0.998 (see Table II).

It seems that in this case the power spectrum approaches the

value 1 more than the DFA method, although we should ver-

ify this for a more ample choice of matrix dimensions. In

Fig. 4c and 5c we show the averaged results for the DFA and

the averaged power spectrum calculations, respectively, for

the energy fluctuations in the GOE spectrum with 347 levels

using an ensemble with 25 sets.

Rev. Mex. F´ ıs. S 54 (3) (2008) 48–55

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E. LANDA, I. MORALES, C. HERN´ANDEZ, V. VEL´AZQUEZ, J.C. L´OPEZ VIEYRA, AND A. FRANK

5.4. GDE

Finally, andforthesakeofcompleteness, weappliedtheDFA

method to the case of the integrable GDE random ensem-

ble. Again we used the same dimension as the case of the

subspace Jπ= 0+. The energy fluctuations represented by

the time series δnand its integrated form ψnare shown in

Fig. 2d and 3d, respectively. In this case, the uncorrelated

nature of the energy fluctuations is noticeable in those fig-

ures. The self similarity parameter was calculated by an av-

eraging procedure over the DFA results (Fig. 4d), and it was

found to be α = 1.34. This value has a deviation of ∼20%

from the expected value of α = 3/2 (corresponding to un-

correlated time series), although for larger dimensions, e.g.,

for N = 1000, the value for the α parameter was α = 1.40,

which is closer to the expected value. On the other hand,

the scaling exponent in the power spectrum was found to be

β = 1.60, which also deviates ∼ 20% from the expected

value of β = 2 for the case of Poisson distributed data. For

a larger dimensionality N = 1000, the value for the β pa-

rameter was β = 1.80, approaching the expected value. The

reason for the above mentioned seems to be due to the poly-

nomial unfolding used in the present study. We must stress

the fact that present calculations should be considered as pre-

liminary.

6.Conclusions

The DFA technique has been applied to the nuclear spectrum

of48Ca for different Jπstates obtained with realistic, and

random Shell Model calculations, to study the scaling prop-

erties of the energy fluctuations around the regular (equally

spaced) spectrum.It was shown that the energy fluctua-

tions in the spectrum of48Ca defined by the stochastic se-

quence (6) exhibit non-trivial scale invariance corresponding

to a critical value of the self similarity parameter α ? 1,

for which the associated autocorrelation function is statisti-

cally scale independent. This result is in agreement with the

corresponding power spectrum calculations for which a sta-

tistical power-law behavior ∼ 1/f is observed. This scaling

of the energy fluctuations was observed in both, random shell

model (TBRE) calculations, as well as in shell model calcu-

lations with realistic interactions. However, the DFA results

appear to be more robust than the power spectrum calcula-

tions. The linear scaling showed in Figs. 4a-4d manifest this

fact. In contrast, the power spectrum calculations depicted in

Figs. 5a-5d display a rather large spread in the Fourier ampli-

tudes, specially in the case of shell model calculations with

a realistic interaction, giving rise to results which are less

transparent. In other cases we require further manipulation

of the data, such as averaging over several calculations. The

DFA method confirms almost perfect non-trivial statistical

scaleinvarianceforhigh-energyfluctuationsin48Ca. Critical

scale invariance was also observed in the case of GOE. Since

this class of scale invariance is observed in several classical

chaotic phenomena, as well as in phase transitional critical

points (logistic maps, geometrical fractals, dripping faucet

experiments, etc.), this result suggests a possible underlying

connection between classical and quantum chaos. This is an

open question which we shall continue to investigate.

Acknowledgments

We are grateful to Jose Barea for his many suggestions. We

thank H. Larralde for his valuable comments. This work was

supported in part by PAPIIT-UNAM and Conacyt-Mexico.

i. Here, a regularized spectrum having ?s? =

is assumed.

∞ ?

0

sP(s)ds = 1,

ii. One of the signatures of fractal structure is associated to the

appearance of energy level repulsion in the quantum spectrum,

which leads to the Wigner surmise of RMT.

iii. The correlation lenght ξ is related to the behavior of the corre-

lation function Γ(r). Near a critical point the correlation func-

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vi. For the time being we assume a finite spectrum with N energy

levels.

vii. Equivalently, we can make a fit of the density of states function

ρ(E), since¯ N(E) ≡?E

ix. For a time series δ(i),i = 1,...N, the autocorrelation func-

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was based on the properties of the resulting δn-time series,

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