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VOLUME 79, NUMBER 23 PHYSICAL REVIEW LETTERS 8DECEMBER 1997
Chaos and Spatial Correlations for Dipolar Eigenproblems
Mark I. Stockman
Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303
(Received 30 July 1997)
Spatial-correlation properties of the wave functions (eigenvectors) of a spin-one eigenproblem for
dipole interaction is studied for random geometries of the underlying system. This problem describes,
in particular, polar excitations (“plasmons”) of large clusters. In contrast to Berry’s conjecture of
quantum chaos for massive particles, we have found long-range spatial correlations for wave functions
(eigenvectors). For fractal systems, not only individual eigenvectors are chaotic, but also the amplitudecorrelation
function exhibits an unusual chaotic, “turbulent” behavior that is preserved by ensemble
averaging. For disordered nonfractal systems, the eigenvectors show a mesoscopic delocalization
transition different from the Anderson transition. [S0031-9007(97)04720-0]
PACS numbers: 05.45.+b, 61.43.Hv, 71.45.Gm, 78.20.Bh
Chaotic behavior of large eigenproblem solutions is of
great significance in physics. Its major application, quantum
chaos, has attracted a great deal of attention. Random
matrices with Gaussian statistics have been employed to
describe universal distributions of quantum levels [1,2],
an idea that goes back to Dyson’s description of complex
nuclear spectra [3]. The applicability of the random matrix
theory has been supported on the basis of a functional
nonlinear s model [4]. Various correlation functions are
useful as characteristic of chaos. In particular, parametric
correlations of chaotic eigenstates has been shown to
be universal [5]. This universality has been confirmed
for the Anderson model of disorder, a tight-binding model
with diagonal (on-site) disorder and nearest-neighbor hopping
[5]. Correlation between eigenvector-overlap intensities
and level velocities has proved to be an important
measure of chaotic scarring [6]. A principal feature of
the above-mentioned studies is the absence of long-range
spatial correlations of chaotic wave functions.
In this paper we will be interested in spatial correlations
and localization of chaotic eigenvectors. Such measures
are of importance as signatures of quantum chaos.
A seminal paper by Berry [7] conjectured that the Wigner
density matrix for a nonintegrable chaotic system obeys a
microcanonical distribution. As a result, the spatial correlation
function behaves as the Bessel function j 1 kr,
resulting in a power-law decay ~1r. The absence of
long-range spatial correlation in chaotic systems with time
reversal (without a magnetic field) has been confirmed
analytically by Fal’co and Efetov using the supersymmetric
s model [8]. Note that in the presence of
an intermediate-strength magnetic field (that is, in the
crossover region between orthogonal and unitary classes)
a weak long-range spatial correlation is predicted [8].
We consider a tight-binding eigenproblem on a discrete
set of coordinates r i (i 1,...,N) for a vector (spin 1)
particle with dipole interaction (hopping amplitude) W,
W 2 w n jn 0,
(1a)
W ia,jb
Ω
r
2
ij d ab 2 3r ij a r ij b r 25
ij , i fi j ,
0, i j ,
(1b)
where jn and w n (n 1, . . . , 3N) are eigenvectors and
eigenvalues, Greek letters in subscripts denote vector indices
(a, b,... x,y,z), and r ij r i 2 r j . Principal
features of this interaction are its infinite range and singularity
at r ij ! 0. These features, spin 1, and nondiagonality
of disorder distinguish Eq. (1) from the Anderson
model.
We show below that the solutions of this eigenproblem
contain strong long-range spatial correlations in chaotic
eigenvectors. We consider spatial disorder of the positions
r i of two types: fractal and nonfractal. Note that without
the spatial disorder the eigenvectors for the long-range interaction
would be trivially smooth. We show that for a
fractal set r i the amplitude correlations are not only long
ranged and strong, but they are also “turbulent,” changing
chaotically with distance and frequency. At the same time,
this randomness is deterministic; i.e., it is characteristic of
the given class of clusters, and it survives ensemble averaging
over different clusters in that class. In a sense, the
chaos found is stronger that that predicted by Berry [7] and
studied, e.g., in Refs. [5,6,8], because not only individual
wave functions but also their correlators are chaotic.
This eigenproblem describes, in particular, eigenstates
of a dipolar excitation (plasmon) on a cluster consisting
of polarizable particles positioned at points r i . The amplitude
of the electric field of an nth eigenmode at ith
particle with polarization a is iajn. The corresponding
Green’s function is G ia,jb w P niajnjbjnw 2
w n 21 . The Green’s function determines, in particular,
the linear response function of the cluster, a ab r, X
2 P n G ia,jb X 1 i´dr 2 r ij , and the density of eigenmodes,
nX p 21 Im P n,i G ia,ia P n d´X 2 w n ,
where X 1 i´ a0 21 , a 0 being the complex polarizability
of a particle, and d´X p 21´X2 1´2 21 . The
relation X Re a0 21 v determines the electromagneticfield
frequency v in terms of the spectral parameter X.
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VOLUME 79, NUMBER 23 PHYSICAL REVIEW LETTERS 8DECEMBER 1997
The imaginary part of the polarizability determines the
dynamic form factor S ab r,X, i.e., the correlation function
of amplitudes at two points separated by a distance r
for eigenvalues w n in the vicinity of X. Namely,
S ab r, X 1 p Im a abX,r
* X
iajnjbjnd´X 2 w n
n,i,j +
3dr2r ij , (2)
where ··· denotes the ensemble average over different
random realizations of the system. The same function
also defines the correlation of the equilibrium fluctuations
of the polarization according to the fluctuation-dissipation
theorem [9]. S ab r,X is the vector counterpart of the
correlation function studied in Ref. [7].
Another useful measure is the second-order pair correlation
function C, i.e., the correlator of probabilities (or
field intensities * for the corresponding classical problem), +
X
Cr, X iajn 2 jbjn 2 d´X 2 w n dr 2 r ij .
n,i,j
(3)
The irreducible second-order correlation function is defined
as
* +
Cr, X X
˜Cr, X 2 I i X I j X dr 2 r ij , (4)
nX
i,j
where the probability (intensity) at a given site i for eigenvalues
w n X is introduced as I i X P niajn 2 3
d´X 2 w n nX. Different from Eq. (3), the correlation
function (4) separates correlation of random fluctuations
from variations of expectation values.
The integral irreducible correlation factor of the quantum
probabilities (or classical intensities) is
K p,q X P p P q X P p X P q X 2 1, (5)
where the moments of probability are defined as
P p P iiajn 2p , and as above, ··· X P n··· 3
d´X 2 w n nX, an average over a narrow spectral
interval of w n X for a given realization of the system.
Note that useful information is obtained only for p, q $ 2
due to normalization. The correlation factor (5) is similar
to the factor studied in Ref. [8]. Asymptotically for large
systems (N ! `), it characterizes long-range spatial
correlations in the eigenvectors.
We have performed numerical computations using two
types of random sets r i of size N 1500: random
lattice gas (RLG) and cluster-cluster aggregates (CCA).
Fractal CCA sets (Hausdorff dimensionality D 1.75)
have been generated on a cubic 3D lattice with unit
period according to Refs. [10,11]. The RLG sets have
been obtained by randomly positioning N points within
a sphere. The density of RLG sets was adjusted to yield
their gyration radius equal to that of the CCA sets. Typically,
the ensemble averaging has been carried out over
300 independent realizations of each type of the sets r i .
The results for the dynamic form factor Sr, X
1
3 S bbr, X [Eq. (2)] for RLG sets and X , 0 are presented
in Fig. 1 (a similar picture is obtained for X . 0).
We see that for extremely large eigenvalues (jXj * 1), the
form factor Sr, X has a quasirandom structure that is reproducible
under the ensemble averaging. In the middle of
the spectral region (0.003 & jXj & 1), the form factor is
dominated by a few branches of excitation. The strongest
one is marked on the lower panel by a solid white line
X 22r 23 , which is the corresponding branch of binary
approximation [12] (where eigenmodes consist of two “hot
spots”) for X , 0, given by
Sr, X drnX 1 fr
3 dX 1 2r 23 2dX22r 23
12dX2r 23 22dX1r 23 , (6)
where fr is a smooth distribution function of the
distance r ij , fr Ndr 2 r ij . The weaker feature,
marked by the dotted line X 21 1 p 57 r 23 , is due
FIG. 1. Dynamic form factor Sr, X for RLG (N 1500),
obtained by averaging over an ensemble of 300 individual sets.
A profile on double-logarithmic scale in r and X is shown
in the upper panel and the corresponding contour map in
the lower panel. The vertical scale is pseudologarithmic to
show simultaneously positive and negative values of Sr, X.
To obtain it, a small region of the plot for jSr, Xj &
10 24 is removed. The function plotted is logjSr, Xj 1
4 sgnSr, X. The solid and dashed white lines in the lower
panel indicate the bands of binary and ternary correlations.
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VOLUME 79, NUMBER 23 PHYSICAL REVIEW LETTERS 8DECEMBER 1997
to the ternary excitations, i.e., eigenvectors consisting of
three hot spots.
At X 0.003, we see a sharp transition to delocalization,
where the positive-correlation region becomes uniform,
spreading over most of the system. Different from
an Anderson transition, this one is mesoscopic, i.e., dependent
on the geometrical size R c of the system. The
transition occurs when the correlation length becomes
comparable to R c , i.e., at jXj Rc
23 ~ N. We found numerically
that this predicted dependence on N holds very
well for 1500 $ N $ 500. The difference from Anderson
localization is due to the long-range interaction in our
case. As a result, most of the eigenvectors are not propagating
waves, but change from binary or ternary excitations
to delocalized surface plasmons as jXj decreases.
A dramatically different picture is obtained for the
dynamic form factor of the fractal CCA sets, as shown in
Fig. 2. We see a developed pattern of irregular, chaotic
correlations in almost the whole spatial-spectral region
r * 5, jXj & 1. There is very little left of the binaryternary
branches, seen so well for RLG. The “landscape”
observed in Fig. 2 well deserves the name of a “devil’s
hill,” where narrow regions of positive and negative
correlation are interwoven, resulting in a turbulencelike
FIG. 2.
Same as in Fig. 1, but for CCA sets.
network. This chaotic behavior is indeed a reflection of
the chaos of the individual eigenvectors. However, this
chaos is in a certain sense stronger, because it survives the
averaging over an ensemble of statistically independent
individual sets (clusters). This deterministic quasichaotic
pattern of the form factor is likely to depend on a specific
topology of the fractal set r i .
Interestingly enough, the profile of Sr, X (see Fig. 2)
consists of alternating domains of positive and negative
correlation, with very sharp transitions (almost vertical
edges) between them. At the same time, the magnitudes
of positive and negative domains are close. This domain
structure is due to the singularity of the dipole interaction.
A simple physical explanation of this fact is that
eigenvectors consist of oscillating dipoles, where nextneighbor
dipoles tend to be positioned either along the
direction of the dipoles (the k configuration) or normal to
it (the configuration). Each oscillating dipole induces
in its neighborhood dipoles of the same magnitude with
the same sign (for the configuration), or of the opposite
sign (for the k configuration).
As one can judge from the upper panel in Fig. 2, the
envelope of the form factor very weakly decreases in r,
slower than any power of r. A power-law decay predicted
in Ref. [7] for a conventional quantum problem would
have manifested itself as straight lines of the landscape
parallel to the S, r plane. Such lines can be seen in
Fig. 2 only at extremely large values of X.
To distinguish between fluctuations of phase and amplitude
in the formation of the devil’s hill in Fig. 2, we
will compare it to the second-order correlation function
Cr, X [Eq. (3)] and its irreducible counterpart ˜Cr, X
[Eq. (4)], shown in Fig. 3. We see that these functions
differ dramatically from the dynamic form factor. The
reliefs in Fig. 3 are very smooth, in contrast to Fig. 2.
This implies that the devil’s hill is formed due to spatial
fluctuations of the phases of individual eigenmodes, while
their amplitudes are smooth functions in space-frequency
domain (see the discussion in the previous paragraph).
Comparison of the upper and lower panes in Fig. 3 shows
that the irreducible correlations exist in a limited range of
the eigenvalues, jXj . 10 23 , while outside of this range
they vanish. A physical reason for that may be a trend to
spatial delocalization of the eigenvectors for jXj !0.
An important feature present in Fig. 3 is the long
range of the correlation. The correlation decay in r is
indeed weaker than any power of r. The lower panel
indicates that the irreducible correlation is likely to behave
logarithmically with r. The spatial extension of the
eigenvectors is determined by typical radius R c of the
system, which is characteristic of the inhomogeneous
localization pattern [13]. Consequently, the correlators
obey scaling Cr, X C rR c , X and similarly for
˜Cr, X. We have checked numerically that this scaling
holds very well. All these features are certainly due to the
long range of the dipole-interaction amplitude.
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VOLUME 79, NUMBER 23 PHYSICAL REVIEW LETTERS 8DECEMBER 1997
FIG. 4. The correlation factor K p,p X [Eq. (5)] for p
2, 3, 4, and 5 as a function of X for a 300-member ensemble
of N 1500 CCA sets.
FIG. 3. Second-order correlation function Cr, X in triplelogarithmic
scale (a) and the corresponding irreducible correlator
˜Cr, X in double-logarithmic scale (b) calculated for an
ensemble of 300 CCA sets of N 1500 points each.
To emphasize the long range of the spatial correlations,
we present in Fig. 4 the correlation factor K p,p X for
p 2, 3, 4, and 5. For electronic quantum-mechanical
problems, K 2,2 X is nonzero only in the presence of an
intermediate magnetic field, where it is 0.1 [8]. As we
see, in our case, the correlator is not small in most of the
spectral region, where K 2,2 X 1.
An interesting question is whether the observed longrange
chaotic correlations are due to the specific tensor
form or only to the long range of the dipole interaction.
To investigate this, we have carried out modeling for a
scalar counterpart of Eq. (1b), W ia,jb rij 23 d ab . The result
is that this scalar interaction also yields chaotic correlations,
albeit in a narrower spatial range of r * 15.
Thus fractality of the set and long range of the hopping
amplitude are qualitatively more important than the specific
tensor structure of the dipole-dipole interaction.
In conclusion, we have found long-range correlations of
the chaotic eigenvectors for the dipole-interaction eigenproblem
on random (fractal or not) sets. For the fractal
systems, the amplitude correlation function (dynamic
form factor) shows an unusual pattern of developed chaotic
correlations, characteristic of the topology of the system,
which is preserved by ensemble averaging. The secondorder
(intensity) correlations are smooth, implying that the
“turbulent” amplitude correlations are due to phase fluctuations.
The correlations are long ranged and strong, in
contrast to eigenproblems for massive particles studied earlier.
For nonfractal disordered systems, the dynamic form
factor reveals a mesoscopic delocalization transition, corresponding
to the appearance of surface plasmons.
I am grateful to S. T. Manson and J. E. Sipe for many
useful discussions. I am thankful to W. H. Nelson for
support and to him and B. D. Thoms for the reading of
the manuscript. I appreciate comments by M. V. Berry on
the results of this paper. I acknowledge with gratitude the
invariable support and help by T. F. George. I thank L. N.
Pandey for providing computer routines for large-matrix
diagonalization. I acknowledge grants from Georgia State
University that allowed the purchase of the state-of-the-art
computer equipment.
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