Scale invariance and criticality in nuclear spectra

Abstract

A Detrended Fluctuation Analysis (DFA) method is applied to investigate the scaling properties of the energy fluctuations in the spectrum of 48Ca 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.

Full text

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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)
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50
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
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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.
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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
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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.
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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-
tion has the Ornstein-Zernike form Γ(r) ∼ r−pe−r/ξwhen
T → Tcrit(see e.g. Ref. 12).
iv. For a given time series δn, the power spectrum is defined as
S(f) = |Ff{δ}|2, where Ff{δ} denotes the component of the
discrete Fourier transform of δ, having frequency f.
v. The Wiener-Khinchin theorem establishes that the power spec-
tral density of a random process is the Fourier transform of
the corresponding autocorrelation function. For a power spec-
trum with a power law behavior, the Wienner-Khinchin theo-
rem implies also a power law behavior of the corresponding
auto-correlation function.
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-
tion is defined as C(τ) =?N
was based on the properties of the resulting δn-time series,
being better for the minimal polynomial degree leading to a
time series oscillating around δ = 0 (when not properly done,
a straightforward high-degree polynomial fit can even lead to
a time series with residual linear tendencies, i.e. oscillating
around a non-horizontal straight line).
1. M.L. Mehta, Random Matrices, 2nd ed. (Academic Press, New
York London, 1991).
2. E.P. Wigner, Results and theory of resonance absorption, Con-
ference on Neutron Physics by Time-of-Flight, Gatlinburg,
Tennessee, 1956.
−∞dE?¯ ρ(E?).
viii. First order DFA.
k=1δ(k)δ(k + τ).
x. The criterion used to choose the degree of the polynomial fit
Rev. Mex. F´ ıs. S 54 (3) (2008) 48–55

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SCALE INVARIANCE AND CRITICALITY IN NUCLEAR SPECTRA
55
3. O. Bohigas, M.-J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52
(1984) 1.
4. M.V. Berry and M. Tabor, Level clustering in the regular spec-
trum, Proc. R. Soc. London A. 356 (1977) 375.
5. T.A. Brody Lett. Nuovo Cimiento 7 (1973) 482.
6. P. Chau, Huu-Tai1, N.A. Smirnova, and P. Van Isacker, Gen-
eralized Wigner surmise for (2 × 2) random matrices, J. Phys.
A35 (2002) L199.
7. P. Cvitanovic, et al., Classical and Quantum Chaos, e-book
(2002).
8. M.C. Gutzwiller, Periodic orbits and classical cuantization
conditions, Jour. Math. Phys. 12 (1971) 343; M.C. Gutzwiller,
Chaos in Classical and Quantum Mechanics (Springer, New
York, 1990).
9. T.P.J. Penna, P.M.C. De Oliveira, J.C. Sartorelli, W.M.
Goncalves, and R.D. Pinto, Long-range anticorrelations and
non-Gaussian behavior of a leaky faucet Phys. Rev. E 52 (1995)
R2168.
10. A.L. Goldberger, D.R. Rigney, and B.J. West, Chaos and frac-
tals in human physiology. Sci Am 262 (1990) 42.
11. A. Rela˜ no, J.M.G. G´ omez, R.A. Molina, J. Retamosa, and E.
Faleiro, Phys. Rev. Lett 89 (2002) 244102; Phys. Rev. E 66
(2002) 036209; E. Faleiro, J.M.G. G´ omez, R.A. Molina, L.
Mu˜ noz, A. Rela˜ no, and J. Retamosa, Phys. Rev. Lett 93 (2004)
244101.
12. K. Huang, Statistical Mechanics, Second Edition (John-Wiley
and Sons, 1987).
13. J. Flores, M. Horoi, M. Muller, and T.H. Seligman, Phys. Rev.
E. 63 (2001) 026204.
14. E. Landa et al. (to be published).
15. C.-K. Peng, et. al., Mosaic organization of DNA nucleotides,
Phys. Rev. E 49 (1994) 1685; C-K Peng, S. Havlin, H.E. Stan-
ley, and A.L. Goldberger, Chaos 5 (1995) 82.
16. F.J. Dyson, J. Math. Phys. V. 3 (1962) 140; 157; 166; 1199.
17. M.S. Santhanam, N. Jayendra, Bandyopadhyay, and Dilip An-
gom, Phys. Rev. E 73 (2006) 015201.
18. C. Heneghan and G. McDarby, Phys. Rev. E 62 (2000) 6103.
19. E.W. Montroll, and M.F. Shlesinger, The wonderful world
of random walks. In: Nonequilibrium Phenomena II. From
StochasticstoHydrodynamics. LebowitzJL,Montroll EW,eds.
(Amsterdam: North-Holland, 1984) p. 1.
20. A.L. Goldberger, et al. Physionet: Components of a New
Research Resource for Complex Physiologic Signals. Cir-
culation 101(23):e215-e220 [Circulation Electronic Pages;
http://circ.ahajournals.org/cgi/content/full/101/23/e215]; 2000
(June 13).
21. T.T.S. Kuo and G.E. Brown, Nucl. Phys. A 114 (1968) 241.
22. J.B. French and S.S.M. Wong, Phys. Lett. B 23 (1970) 449;
S.S.M. Wong and J.B. French, Nucl. Phys. A 198 (1972) 188;
O. Bohigas and Flores, J. Phys. Lett. B 34 (1971) 261.
23. T. Papenbrock and H.A. Weidenm¨ uller, Rev. Mod. Phys. 79
(2007) 997.
24. E. Caurier, computer code ANTOINE, CRN, Strasbourg (2000);
E.Caurier, A.P. Zuker, and A. Poves: in Nuclear Structure of
Light Nuclei far from Stability. Experiment ad Theory, Proceed-
ings of the Workshop, Obrnai, Ed. G. Klotz (CRN, Strasbourg,
1989).
Rev. Mex. F´ ıs. S 54 (3) (2008) 48–55