Download Wave functions in the Anderson model and in the quantum

Ann. Phys. (Leipzig) 7 (1998) 5Ð6, 400Ð405
Wave functions in the Anderson model and
in the quantum percolation model: a comparison
Jan W. Kantelhardt and Armin Bunde
Institut fur Theoretische Physik, Justus-Liebig-Universitat Giessen, D-35392 Giessen,
Germany
Received 6 October 1998, accepted 9 October 1998 by B. Kramer
Abstract. We compare numerically the localization behavior of electronic eigenfunctions
in the Anderson model and on self-similar percolation clusters at criticality. We ÿnd that
the distributions of the local wave function amplitudes j j at ÿxed distances from the localization center are very similar for both models. The amplitude distributions are well
approximated by log-normal ÿts, which seem to become exact at large distances. From
the distributions, we can calculate analytically the behavior of the averages at suþciently
large distances. We observe two diýerent localization regimes. In the ÿrst regime, at intermediate distances from the localization center, we ÿnd stretched exponential localization
('sublocalization'), lnhj ji ÿ þ dÿ , with eýective localization exponents ÿ 1. In the second regime, for very large , the averages strongly depend on the number of conÿgurations
, and superlocalization ( ÿ 1) is observed, converging to simple exponential behavior
asymptotically as expected. The crossover from the intermediate to the asymptotic regime
depends logarithmically on the number of conÿgurations.
ÿ
ÿ
r
d
<
r
N
d
>
Keywords: Quantum percolation; Anderson model; Amplitude distributions
1 Introduction
The percolation model [1] and the Anderson model [2] are the two standard models
for disordered solids. They diÿer in the type of disorder, but for both models the wave
functions of quantum particles are localized in ÿ 2 or for suþciently large disorder in
2. In this paper, we compare the statistics of the wave functions for both models
by investigating the distribution of amplitudes at ýxed distance from the localization
center in the regime of strong localization. We show that the average decay of the
wave functions can directly be calculated from these amplitude distributions.
d
d >
2 Models and averaging procedure
We consider the Schrodinger equation in tight-binding approximation for the wave
function of a quantum particle on regular lattices. The coeþcients for each lattice
site satisfy the tight-binding equation
X
=
+
(1)
+
+
ÿn
n
E ÿn
þn ÿn
Vn;n
ÿ
ÿ ÿn
ÿ;
401
J. W. Kantelhardt, A. Bunde: Wave functions
where the sum runs over all nearest neighbor sites + of site . j j2 is the
probability density on site for the wave function with energy . The hopping
terms
+ are constant for nearest neighbor cluster sites and zero otherwise; for
simplicity we take = 1 as energy unit. For the diagonal terms of the Hamiltonian
we consider two diÿerent kinds of disorder:
In the Anderson model (AM) all sites of the underlying lattice have ýnite . The
values are randomly chosen from the interval [þ 2 2], where is the disorder
parameter (diagonal disorder) [2]. Here, we consider the Anderson model for a linear
chain and on the square lattice. In these cases the wave functions are known to
be localized. The statistics of wave function intensities for all lattice sites has been
calculated in [3].
In the quantum percolation (QP) model we have structural disorder. The sites of the
underlying lattice are randomly chosen to have either = 0 (\occupied site") or =
1 (\empty site") [4, 5]. On empty sites the amplitude j j2 must always be zero; thus
these sites (and their bonds) can be dropped from the lattice. At small concentrations
of occupied sites only ýnite clusters of occupied nearest neighbor sites exist, while
for large there are only small holes in the cluster of occupied sites [1, 6]. Here
we consider the incipient inýnite percolation cluster at the threshold concentration
c (\critical concentration") where for the ýrst time a cluster of occupied nearest
neighbor sites reaches opposite boundaries of the system. The values of c are known
from numerical simulations: c ' 0 592746 for the square lattice, c ' 0 3116 for the
simple cubic lattice, or analytically: c = 0 5 for percolation clusters on the Cayley
tree with coordination number = 3 [1, 6]. At c the wave functions are always
strongly localized due to the strong structural disorder. On percolation clusters two
diÿerent distances can be deýned between two sites: The topological (or chemical)
distance is the shortest path distance on the cluster and is always larger than the
spatial (or Euclidian) distance measured directly without looking at the structure
of the cluster.
n
ý
n
Vn;n
ÿn
E
n
ÿ
V
þn
þn
w= ; w=
w
þn
þn
ÿn
p
p
p
p
p
:
p
p
:
:
z
p
`
r
3 The amplitude distributions
We are interested in the distribution of the amplitudes j j for the eigenfunctions of
Eq. (1). Because of the disorder, the eigenvalue equation has to be solved numerically.
We employed a Lanczos algorithm for the AM in = 2 and for QP at criticality in
= 2 and = 3. For the AM in = 1 we used the method described in [7], while an
improved version of this method described in [8] was used for QP on the Cayley tree.
Since we are interested in the decay of the localized wave functions, we investigate the
distribution of the amplitudes j j at ýxed distances from the localization center. By
deýnition, the localization center is that site, where the maximum of the amplitudes
occurs. Our results for the amplitude distribution function are shown in Fig. 1. The
histogram function (j j ) is the probability of ýnding the amplitude j j at distance
from the localization center. Fig. 1(a) and (b) show the results for the AM. For QP,
the distributions for given topological distances or given spatial distances have to
be distinguished, and we show both for QP in = 2 in Fig. 1(c) and (d). For QP in
= 3 and on the Cayley tree we show only the results in -space (in Fig. 1(e) and
(f)).
We ýnd that the histograms of the amplitudes j j at ýxed distance or (for QP)
ÿ
d
d
d
d
ÿ
H
ÿ ;r
ÿ
r
`
r
d
d
`
ÿ
r
`
402
Ann. Phys. (Leipzig) 7 (1998) 5Ð6
1.0
0.8
(a) AM
in d = 1
H ( ψ , r ) ⋅ 0.6
1.0
p = pc,
λ = 10.6, 0.8
σ = 2.6
(d) QP
in d = 2,
l-space
w = 1.0,
λ = 105,
σ = 2.1
0.6
πσ r λ 0.4
0.4
0.2
0.2
0.0
1.0
0.0
1.0
(e) QP
w = 8.5,
λ = 2.9...6.6, in d = 3,
σ = 1.2
l-space
(b) AM
in d = 2
0.8
H ( ψ , r ) ⋅ 0.6
p = pc,
λ = 7, 0.8
σ=3
0.6
πσ r λ 0.4
0.4
0.2
0.2
0.0
1.0
0.0
1.0
0.8
H ( ψ , r ) ⋅ 0.6
p = pc, (f) QP
λ = 5.4, on the
σ = 2.6
(c) QP
in d = 2,
r-space
p = pc,
λ = 5.9, 0.8
σ = 2.9
Cayley
tree
πσ r λ 0.4
0.6
0.4
0.2
0.2
0.0
-2
-1
0
1
2
3 -2
-1
0
1
− ln ψ − r λ
− ln ψ − l λ
σr λ
σl λ
0.0
3
2
Fig. 1 Scaled distributions (j j ), respectively (j j ) of the amplitudes at ÿxed distances respectively from the localization center for diýerent models: AM in (a) = 1
and (b) = 2; QP in = 2 in (c) -space and (d) -space; -space for QP (e) in = 3 and
(f) on the Cayley tree. The data for diýerent ÿxed distances and is scaled with the two
parameters and given in the graphs, so that they fall onto a single curve for diýerent
values of (or ). The continuous black line in each graph is the log-normal distribution
function, (j j ), Eq. (2) (or (j j )).
H
r
ÿ ;r
H
ÿ ;`
`
d
d
d
`
r
`
d
r
þ
r
H
ý
`
ÿ ;r
H
ÿ ;`
`
403
J. W. Kantelhardt, A. Bunde: Wave functions
from the localization center surprisingly well obey a log-normal distribution,
ÿ (ln j j + )2 þ
1
(j j ) = p
exp þ
H
ÿ
ÿ ;r
r=ú
üû r=ú
(2)
;
û r=ú
Note that the parameter is the localization length, since the amplitude exp(þ )
occurs with the largest probability at distance from the center of localization according to Eq. (2). The distributions are less broad by a factor of approximately two
for the AM compared to the QP model (for the same dimension), but their shapes
are the same.
On percolation clusters, the distribution of the amplitudes for ýxed distance from
the center of localization is not exactly log-normal, as can be seen in Fig. 1(c). But the
distribution of the relevant large amplitudes j j can surprisingly well be approximated
by a log-normal ýt. The continuous line in Fig. 1(c) represents a ýt to the left part
of the histogram by the (renormalized) log-normal distribution Eq. (2).
ú
r=ú
r
r
ÿ
4 The averaged amplitudes
Now we are interested in a quantitative description of the way the average values of j j
decay with increasing distance from the localization center. The necessary averaging
procedure consists of three steps: In the ýrst step we average, for each eigenfunction
=1 2
, the values of j j2 at ýxed distance from the localization center. The
resulting function (2 ) ( ) characterizes the spatial decrease of our quantity of interest
in the considered th conýguration. In the second step, for obtainingP
the mean spatial
decrease, we average ( ) ( ) over conýgurations, h ( )i = 1 =1 ( ) ( ). If
the system is not selfaveraging (which we ýnd is the case here), the resulting values
for h ( )i will depend on the special set of conýgurations considered and will
üuctuate from set to set. In this case, to obtain the typical value of h ( )i , we are
led, in the third step, to the logarithmic average over many sets of conýgurations,
i. e. we average lnh ( )i over many sets of eigenfunctions [8, 9]. The resulting
function û ( ) ý exphlnh ( )i i can be considered as the typical average. In the
following, we shall discuss exclusively this \typical average over conýgurations".
Since we can very well approximate the distribution of dominating amplitudes j j at
distances ü (or ü for QP) by Eq. (2), we can directly calculate the decay
of the average wave function û ( ) for large by simple integration [8, 10]. We can
also estimate the way û ( ) depends on the number of averaged conýgurations .
In order to describe the localization behavior with an eÿective localization exponent
we make the ansatz
ÿ
r
ù
;
;:::;N
ÿ
ÿ
r
r
þ
ù
ÿ þ
ÿ r
r
N
ÿ r
N
N
þ
N
ÿ þ
r
N
N
ÿ r
N
N
ÿ r
N
N
N r
ÿ r
N
N
ÿ
r
ú
`
ú
N
r
r
N r
N
dý
ln û ( ) û þ( ) ÿ
(3)
where again has to be replaced by for the topological space in QP.
The
calculation of û ( ) from (j j ) via the convolution integral û ( ) =
R 1 analytical
ÿ ( ÿ ) is straightforward, but beyond the scope of this paper; we
(
)
min
refer to [8, 10]. The number of conýguration does only enter through the lower
integration bound min ( ), which becomes zero for small and large . Thus,
there is an intermediate regime for
þ ( ), where û ( ) is independent of
. It can be shown, that the decay of û ( ) is well approximated by Eq. (3) with
N
r
r;N
e
x
H e
r=ú
d
;
`
N
x
r
x
;r
r
H
ÿ ;r
N
dx
N
x
r; N
r
ú < r < r
N
N
r
N
N
N
r
r
404
Ann. Phys. (Leipzig) 7 (1998) 5Ð6
ΨN( r )
ΨN(" )
-1
10
10
-10
-1
-10
10
10
(a)
-100
10
(b)
-1
10
0
10
1
r/λ10
10
2
-100
0
10
1
2
10
"/λ
10
10
Fig. 2 Averaged amplitudes üN ( ), respectively üN ( ) for wave functions (a) in the Anderson model in = 1 and (b) on critical percolation clusters on the Cayley tree. The curves
for several ranging from 1 to 65000 are shown. The dashed lines correspond to simple
exponential localization ( ÿ = 1) and are shown for comparison.
r
`
d
N
d
an eÿective localization exponent ú 0 6 in this regime. Since the exponent is signiýcantly smaller than one, the decay is slower than exponential. Hence, we can call
the intermediate regime (
þ ( )) the regime of \sublocalization". If a large
number of eigenfunctions is averaged, the decay is dominated by this sublocalization
regime. The crossover length þ ( ) can be shown to depend logarithmically on the
number of averaged conýgurations , þ ( ) ú ln [8, 10].
For large or small (
þ ( )), the lower bound in the convolution integral does
depend on . In this case the arithmetically averaged eigenfunctions û ( ) also
depend on , i. e. selfaveraging breaks down. In this regime the decay of û ( ) is
well approximated by Eq. (3) with an eÿective localization exponent
1, which
depends on and and drops to 1 asymptotically for ! 1 [8, 10].
Since the amplitude distributions (j j ) (for the AM) and (j j ) (for QP) are
both well approximated by Eq. (2), the two localization regimes with the -dependent
crossover þ ( ) can be found for the AM as well as for QP. This is illustrated in Fig. 2,
where we show û ( ) for the AM in = 1 and û ( ) for QP on the Cayley tree for
various .
In conclusion, we have compared the distributions of the amplitudes of wave functions in the Anderson model and the quantum percolation model at criticality. We
ýnd that the distributions are well described by a log-normal function, which has
the same shape for both models and in several dimensions, and depends only on two
parameters. The averaged amplitudes also decay in a very similar fashion, that is
dominated by a pronounced regime of sublocalization for the relevant distances from
the localization center.
dý
ú < r < r
r
:
N
N
N
r
N
r > r
r
N
úû
N
N
N
N
N
N
dý >
r
N
r
H
ÿ ;r
H
ÿ ;`
N
r
N
N
N
r
r
d
N
`
r
J. W. Kantelhardt, A. Bunde: Wave functions
405
We thank the Deutsche Forschungsgemeinschaft for ÿnancial support.
References
[1] A. Bunde, S. Havlin (Eds.), Fractals and Disordered Systems, 2nd ed., Springer Verlag,
1996; D. Stauýer, A. Aharony, Introduction to Percolation Theory, 2nd ed., Taylor &
Francis, 1992
[2] B. Kramer, A. MacKinnon, Rep. Prog. Phys. 56 (1993) 1469
[3] K. Muller, B. Mehlig, F. Milde, M. Schreiber, Phys. Rev. Lett. 78 (1997) 215
[4] S. Kirkpatrick, T. P. Eggarter, Phys. Rev. B 6 (1972) 3598
[5] A. Mookerjee, I. Dasgupta, T. Saha, Int. J. Mod. Phys. B 9 (1995) 2989
[6] A. Bunde, J. W. Kantelhardt, L. Schweitzer, this volume
[7] H. E. Roman, C. Wiecko, Z. Phys. B 62 (1986) 163
[8] J. W. Kantelhardt, A. Bunde, Phys. Rev. E 56 (1997) 6693
[9] A. Bunde, J. Drager, Phys. Rev. E 52 (1995) 53
[10] A. Bunde, J. Drager, J. W. Kantelhardt, Proceedings of the XI. Max-Born-Symposium
on Anomalous Diýusion, Springer Verlag, in press (1998)