1D Anderson model revisited: Band center anomaly for correlated disorder
We study the band-center anomaly in the one-dimensional Anderson model with the disorder characterized by short-range positive correlations. Using the Hamiltonian map approach, we obtain analytical expressions for the localization length and the invariant measure of the phase variable. The analytica...
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irk-123456789-1293792018-01-20T03:03:46Z 1D Anderson model revisited: Band center anomaly for correlated disorder Herrera-González, I.F. Izrailev, F.M. Makarov, N.M. Tessieri, L. К 100-летию со дня рождения И.М. Лифшица We study the band-center anomaly in the one-dimensional Anderson model with the disorder characterized by short-range positive correlations. Using the Hamiltonian map approach, we obtain analytical expressions for the localization length and the invariant measure of the phase variable. The analytical expressions are complemented by numerical data. 2017 Article 1D Anderson model revisited: Band center anomaly for correlated disorder / I.F. Herrera-González, F.M. Izrailev, N.M. Makarov, L. Tessieri // Физика низких температур. — 2017. — Т. 43, № 2. — С. 347-352. — Бібліогр.: 20 назв. — англ. 0132-6414 PACS: 71.23.An, 72.15.Rn, 05.40.–a, 42.25.Dd http://dspace.nbuv.gov.ua/handle/123456789/129379 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We study the band-center anomaly in the one-dimensional Anderson model with the disorder characterized by short-range positive correlations. Using the Hamiltonian map approach, we obtain analytical expressions for the localization length and the invariant measure of the phase variable. The analytical expressions are complemented by numerical data. |
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Herrera-González, I.F. Izrailev, F.M. Makarov, N.M. Tessieri, L. |
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Herrera-González, I.F. Izrailev, F.M. Makarov, N.M. Tessieri, L. |
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Herrera-González, I.F. |
| title |
1D Anderson model revisited: Band center anomaly for correlated disorder |
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1D Anderson model revisited: Band center anomaly for correlated disorder |
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1D Anderson model revisited: Band center anomaly for correlated disorder |
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1D Anderson model revisited: Band center anomaly for correlated disorder |
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1D Anderson model revisited: Band center anomaly for correlated disorder |
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1d anderson model revisited: band center anomaly for correlated disorder |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 100-летию со дня рождения И.М. Лифшица |
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1D Anderson model revisited: Band center anomaly for correlated disorder / I.F. Herrera-González, F.M. Izrailev, N.M. Makarov, L. Tessieri // Физика низких температур. — 2017. — Т. 43, № 2. — С. 347-352. — Бібліогр.: 20 назв. — англ. |
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Физика низких температур |
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2025-07-09T11:16:06Z |
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| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2, pp. 347–352
1D Anderson model revisited: Band center anomaly
for correlated disorder
I.F. Herrera-González1, F.M. Izrailev1, N.M. Makarov2, and L. Tessieri3
1Instituto de Fisica, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, México
2Instituto de Ciencias, Benemérita Universidad Autónoma de Puebla, Edificio IC10, Ciudad Universitaria,
Col. San Manuel, Puebla, Pue. 72570, México
E-mail: makarov.n@gmail.com
3Instituto de Fisica, y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia 58060, México
Received 27 August, 2016, published online December 26, 2016
We study the band-center anomaly in the one-dimensional Anderson model with the disorder characterized
by short-range positive correlations. Using the Hamiltonian map approach, we obtain analytical expressions for
the localization length and the invariant measure of the phase variable. The analytical expressions are comple-
mented by numerical data.
PACS: 71.23.An Theories and models; localized states;
72.15.Rn Localization effects (Anderson or weak localization);
05.40.–a Fluctuation phenomena, random processes, noise, and Brownian motion;
42.25.Dd Wave propagation in random media.
Keywords: Anderson localization, band-center anomaly.
1. Introduction
More than half a century has passed since the celebrated
Anderson model (A-model) was introduced to prove the
absence of diffusion of electrons in infinite lattices with
disordered potentials [1]. To date, this model has been ex-
tensively studied due to its various non-trivial physical
properties. First, it was proved that all electronic eigen-
states in the 1D (one-dimensional) model are exponentially
localized in infinite samples [2]. The characteristic length
scale of such a spatial localization is called the localization
length locL . Second, it was shown [3] that the knowledge
of this length allows one to predict all transport properties
for finite samples of size N , for any value of the key pa-
rameter which is the ratio loc /L N (see, also, Ref. 4 and
references therein). This result is known as the single-
parameter scaling (SPS) and for a long time it was be-
lieved to be true for any eigenstate energy E within the
allowed band | | < 2E , with the exception of energies close
to the band edges, = 2E ± (for the normalized energy).
The first analytical expression for locL as a function of
the energy E was given by Thouless [5] under the condi-
tions of weak and uncorrelated disorder. However, shortly
afterwards it was numerically found [6] that the Thouless
expression for the discrete A-model is not correct for ener-
gies close to the band center, 0E . Although the discrep-
ancy was quite small, a theoretical explanation was not
found until the analytical studies published in [7,8]. It was
established that the standard perturbation theory fails at the
band center because it is based on the assumption of a non-
degenerate spectrum of the Perron–Frobenius operator. An
approximate expression for the localization length was then
derived in Ref. 7 with the use of the degenerate perturbation
theory. Later, with the use of a different approach, the band
center anomaly was analytically resolved in Ref. 8.
It was understood that the mechanism of the band cen-
ter anomaly can be ascribed to the resonance that emerges
for = 0E in the absence of disorder. The physical origin of
this effect is that the shift µ of the phase of the wave func-
tion turns out to be /2π for every lattice step. Therefore,
even for weak disorder, the phases remain strongly corre-
lated. Later on, the anomalous behavior of the localization
length in a neighborhood of the band center was studied in
[9]. Finally, a complete solution for the localization length
in a whole vicinity of the band center was obtained in
Ref. 10. This solution was derived with the Hamiltonian
map approach, which is based on the mathematical corre-
spondence between the 1D Anderson model and a classical
parametric oscillator [11,12]. Further theoretical contribu-
tions are given in Ref. 8, were it was shown that similar
resonances occur for = 1E ± , as well as for all other reso-
© I.F. Herrera-González, F.M. Izrailev, N.M. Makarov, and L. Tessieri, 2017
I.F. Herrera-González, F.M. Izrailev, N.M. Makarov, and L. Tessieri
nant values of the energy, = 2cos( )E rπ , with r a rational
number. However, only the resonances for = 1/2;1; 2r affect
directly the localization length. As for other anomalies, one
can expect that they influence the higher-order terms in the
expansion of the localization length with respect to disor-
der. Expressions for the localization length at the band
edges ( = 1,2r ) can be found in Refs. 9 (see also [4]).
A second phase of research on the band-center anomaly
began when the anomalies in the 1D Anderson model were
related to the question of the validity of the single parame-
ter scaling. The SPS theory, originally proposed in Ref. 13,
provides a fundamental theoretical tool for the understand-
ing of Anderson localization. For this reason an intense
(and still ongoing) debate was sparked when it was shown
that the SPS theory fails close to the band edges [14] and
to the band center [15]. Since then, the nature of the anom-
alies in the 1D Anderson model has been analyzed with an
eye to the foundations of the SPS theory [16].
More recently, the analysis of the band center anomaly
has been extended to the case of the 1D Anderson model
with correlated disorder [17,18]. In particular, in Ref. 18 it
was shown how correlations of the disorder can either en-
hance or suppress the anomaly at the band center.
In this paper we apply the general results of Ref. 18 to
the study of an interesting case which was numerically
investigated in Ref. 19. The authors considered the 1D
Anderson model with a specific kind of short-range corre-
lated disorder; their numerical analysis revealed that the
localization length at the band center could behave in un-
expected ways for increasing values of the correlation
length of the disorder. In particular, for weak disorder it
was found that the localization length increases with the
third power of the correlation length. Although the authors
of Ref. 19 could not compute the localization length for
very large values of the correlation length in the weak dis-
order regime, they did it for a relatively strong disorder and
found that increasing the correlation length leads to more
and more extended states at the band center.
Our analysis makes possible to understand the numerical
results reported in Ref. 19. In fact, we were able to derive an
analytic expression for the localization length at the band cen-
ter in the case of weak disorder. The obtained expression re-
produces the observed dependence of the localization length
on the correlation length. In particular our results show that, as
the correlation length is increased, the eigenstates do tend to
be more delocalized at the band center.
We also found that longer correlation lengths tend to
suppress the band center anomaly. Note that, in the context
of the 1D Anderson model with correlated disorder, any
anomaly must be defined as the discrepancy between the
effective localization length and the value predicted by the
IK-formula originally derived in Ref. 20 (see also [4] and
references therein). Incidentally, this suppression of the
band-center anomaly was the reason that stimulated some
of us to conduct the research work published in [18].
The paper is organized as follows. In Sec. 2 we define
the model under study. In Sec. 3 we summarize the main
theoretical results that describe the band center anomaly
when disorder is spatially correlated. In Sec. 4 these gen-
eral results are applied to the model analyzed in Ref. 19.
We draw our conclusions in Sec. 5.
2. Definition of the model
The tight-binding Anderson model has the form of the
discrete stationary Schrödinger equation,
1 1 = .n n n n nE+ −ϑψ +ϑψ + ε ψ ψ (1)
Here nε are random site energies and ϑ is the parameter
standing for the coupling between nearest sites. In this
equation the eigenstates nψ and their energies E are fully
determined by the properties of disorder nε ; for this reason
in what follows we put = 1ϑ .
In order to define the site energies nε , we first generate
a sequence { }nη of identically distributed independent ran-
dom variables with the distribution,
[ 1/2, 1/2]1 if
( ) =
[ 1/2, 1/2]0 if
n
n
n
p
η ∈ −
η η ∉ −
.
The colored noise { }nβ , introduced in Ref. 19, is then ob-
tained by filtering the white noise { }nη with an exponential
weight function,
| |/ c
=
= e .n m l
n n
m
∞
− −
−∞
β η∑ (2)
Finally, the site energies { }nε are specified by the rescaling
of the { }nβ variables,
2
= .n
n
n
β
ε σ
〈β 〉
(3)
Here and in the rest of the paper we use the symbol 〈 〉 to
denote the average over disorder realizations. The parame-
ter σ in Eq. (3) defines the intensity of the disorder and in
this paper we consider the case of weak disorder,
2 1.σ << (4)
It is easy to see that, because = 0n〈η 〉 , one also has
= 0.n〈ε 〉
Taking into account that the { }nη variables are independ-
ent, one can derive the binary correlator of the site energies
from Eqs. (2) and (3). One obtains that
[ ] | |2 cc= 1 | | tanh ( ) e k l
n n l l k −
+〈ε ε 〉 σ +
with
c
c
1= .k
l
The corresponding normalized correlator is
348 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2
1D Anderson model revisited: Band center anomaly for correlated disorder
[ ] | |cc2( ) = = 1 | | tanh ( ) e .k ln n l
n
l l k −+〈ε ε 〉
+
〈ε 〉
(5)
The exponential factor in the binary correlator (5)
shows that the parameter cl , introduced in Eq. (2), is essen-
tially the correlation length for the site energies. The
prefactor preceding the exponential term in Eq. (5), on the
other hand, shows that the random site energies (3), strictly
speaking, are not exponentially correlated. This would be
the case only if the binary correlator (5) were of the form,
| |ce ( ) = e .k ll − (6)
This may seem a pedantic remark; however, as we shall
see, the linear prefactor in the binary correlator (5) plays an
important role in accelerating the suppression of the band
center anomaly for increasing values of cl . It should be
noted that in Ref. 19 the authors did not specify the analyt-
ic form of the binary correlator (5), relying instead on nu-
merical evidence to show that it exhibits a roughly expo-
nential decay for c| |l l>> (see Ref. 19).
3. The Hamiltonian map approach
The Hamiltonian map approach, introduced in [11,12],
represents a useful tool to analyze the electronic states of
the Anderson model. The method relies on the correspond-
ence between the Anderson model (1) and a classical sto-
chastic oscillator,
[ ]
2
2 21= 1 ( ) .
2 2
pH x t+ µ + ξ (7)
In Eq. (7) the symbol ( )tξ represents a succession of delta
kicks of random strengths
( )
=
( ) = .n
n
t t n
∞
−∞
ξ ξ δ −∑
After integrating the dynamical equations of the parametric
oscillator (7) over the period = 1T between two kicks, one
obtains the Hamiltonian map
[ ]
[ ]
1
1
1cos sin sin ,
sin cos cos .
n n n n
n n n n
x x p
p x p
+
+
= µ −µξ µ + µ
µ
= −µ µ +µξ µ + µ
(8)
The correspondence between the models (1) and (7)
emerges clearly if one eliminates the momenta from the
map (8). In this way one obtains the equation
1 1 sin = 2cos .n n n n nx x x x+ −+ +µξ µ µ (9)
Equations (1) and (9) have the same structure; their com-
parison reveals that the parameters of the two models must
obey the identities
= 2cos and = sin .n nE µ ε µξ µ
The dynamics of the oscillator (7) is best studied with
the use of action-angle variables ( , )n nJ θ , which are relat-
ed to the Cartesian coordinates by the relations
2
sin ,
2 cos .
n
n n
n n n
J
x
p J
= θ
µ
= µ θ
In terms of these new variables the map (8) becomes
( )
22 3
1
2
( ) cossin sin
(mod 2 ),
n n n n n n n
o
+θ = θ +µ +µξ θ + µξ θ θ +
+ σ π
(10)
( )2 2
1 = 1 2 sin cos ( ) .sinn n n n n n nJ J+ − µξ θ θ + µξ θ
Note that the map (10) for the angular variable is an ap-
proximation, valid in the weak-disorder case (4). The Lan-
dau symbol in Eq. (10) represents neglected terms of order
higher than the second in the perturbative parameter σ . For
the sake of simplicity, in the rest of this paper we omit the
symbol 2( )o σ ; all identities must be interpreted as correct
within the limits of the second-order approximation in the
disorder strength.
The inverse localization length (or Lyapunov exponent)
loc= 1/Lλ is
1=1
1= ln .lim
N
n
N nnN→∞ −
ψ
λ
ψ∑ (11)
For weak disorder, and away from the band edges, the
Lyapunov exponent (11) can be written in terms of the
action-angle variables as
( ) ( )
( )
2
2= 1 2 cos 2 cos 4
8
sin 2 .
2
n n n
n n
µ
λ 〈ξ 〉 − 〈 θ 〉 + 〈 θ 〉 −
µ
− 〈ξ θ 〉
(12)
The noise-angle correlator sin (2 )n n〈ξ θ 〉 in Eq. (12) can
be evaluated following the method introduced in Ref. 20.
We first define the family of noise-angle correlators
2= e .i n ll nq θ −〈ξ 〉 (13)
After dropping the second-order terms in the θ-map (10),
one can write
2
1 = .sinn n n n+θ θ +µ +µξ θ (14)
With the use of the map (14) one can show that the
correlators (13) obey the recursive relation
22 2 2
1 = e 2 e ( ) .sin
ii nl l n nq q i lθµ
−
+ µ〈ξ 〉〈 θ 〉
After multiplying both sides of this equation by 2 ( 1)ei lµ −
and summing over the index l , one obtains
22 2 2
0
=1
= 2 e sin ( )e .i i lnn n
l
q i l
∞
θ µµ〈ξ 〉〈 θ 〉∑
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 349
I.F. Herrera-González, F.M. Izrailev, N.M. Makarov, and L. Tessieri
The imaginary part of this identity gives
( )
( ) ( ) ( )
( ) ( ) ( )
2
=1
=1
sin 2
2
1 2 cos 2 cos 4 ( )cos 2
2 sin 2 sin 4 ( )sin 2 .
n n n
n n
l
n n
l
l l
l l
∞
∞
µ
〈ξ θ 〉 = − 〈ξ 〉 ×
× − 〈 θ 〉 + 〈 θ 〉 µ +
+ 〈 θ 〉 − 〈 θ 〉 µ
∑
∑
(15)
If the noise-angle correlator (15) is inserted in Eq. (12), the
expression for the Lyapunov exponents becomes
( ) ( ){
( ) ( ) }
2
2 1 2 cos 2 cos 4 ( )
8sin
2 sin 2 sin 4 ( ) .
n n
n n
K
Y
σ
λ = − 〈 θ 〉 + 〈 θ 〉 µ + µ
+ 〈 θ 〉 − 〈 θ 〉 µ
(16)
In Eq. (16) the term,
( )
=1
( ) = 1 2 ( )cos 2
l
K l l
∞
µ + µ∑ (17)
is the power spectrum of the disorder while
( )
=1
( ) = 2 ( )sin 2
l
Y l l
∞
µ µ∑ (18)
is the sine transform of the binary correlator (5).
It is important to stress that Eq. (16) contains averages
which can be evaluated only if one knows the invariant
distribution ( )ρ θ for the angle variable. For most values of
the energy in the Anderson model, the map (10) ensures
that the angle variable has a uniform distribution
1( ) = .
2
ρ θ
π
Making use of this distribution in Eq. (16), one obtains the
IK-formula originally derived in Ref. 20,
2
2= ( ).
8sin ( )
IK Kσ
λ µ
µ
(19)
Equation (19) gives the inverse localization length for the
1D Anderson model with weak correlated disorder. It rep-
resents a generalization of the Thouless formula, to which
it reduces for uncorrelated disorder.
For the case of interest here, however, Eq. (19) cannot
be applied, because the invariant measure is modulated,
not uniform. This is due to the fact that 0E corresponds
to /2µ π and for this value of the µ parameter the angle
map (10) has almost periodic orbits of period 4, which ul-
timately lead to the modulation of ( )ρ θ .
The invariant distribution ( )ρ θ close to the band center
can be obtained with the method introduced in [12] for the
case of uncorrelated disorder and extended in [18] to the
case of correlated disorder. In this approach one first con-
siders the fourth iterate of the map (10) with / 2µ π ; the
continuum limit is then taken and the map is replaced with
a stochastic differential equation for ( )tθ . The invariant
measure for θ is eventually obtained by solving the sta-
tionary Fokker–Planck equation associated to the stochas-
tic differential equation previously derived. The interested
reader can find a detailed explanation of the derivation in
Ref. 18.
After lengthy calculations, one obtains that the invariant
measure at the exact band center is
( ) [ ]2
1 1( ) =
2 4 2 1 cos(4 )
ρ θ
α − α − θK
(20)
with
( /2)= .
(0) ( /2)
K
K K
π
α
+ π
(21)
The corresponding inverse localization length is
( )
( )
2
= (0) (0) .
4 2
K K K
ασ π λ + − α
E
K
(22)
In Eqs. (20) and (22) the symbols ( )kK and ( )kE represent
the complete elliptic integrals of the first and second kinds.
Equations (20) and (22) are the main theoretical results that
will be used below to analyze the Anderson model with the
short-range correlated disorder introduced in Ref. 19.
4. The band center anomaly in the Sales-de Moura
model
When the binary correlator of the site energies in the
Anderson model (1) has the form (5), the power spectrum
of the disorder takes the form
[ ]
3
c
2
c c
( )sinh( ) = .
cosh( ) cosh( ) cos(2 )
k
K
k k
µ
− µ
(23)
From Eqs. (22) and (23), one easily obtains that the inverse
localization length is equal to
[ ]
2
22 1
c c
c 1
( )
= 2 1 cosh ( ) 1 cosh( ) ,
2sinh(2 ) ( )
k k
k
ασ λ + − + α
E
K
(24)
where the parameter (21) assumes the specific value
[ ]
2
c
1
2
c c
( )sinh= .
1 cosh( ) 2 1 cosh ( )
k
k k
α
+ +
(25)
Equation (24) gives the inverse localization length at the
exact band center for the Anderson model (1) with the
Sales-de Moura correlated disorder (3).
One can gain physical insight into the behavior of the
inverse localization length (24) by considering the limit
350 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2
1D Anderson model revisited: Band center anomaly for correlated disorder
cases of a very short and very long correlation length cl . In the
limit c 0l → , i.e., for ck →∞ , the Lyapunov exponent (24)
reduces as expected to the Derrida–Gardner form [8,12],
2 (1/ 2)= 2 1 .
4 (1/ 2)
σ
λ −
E
K
(26)
In the opposite limit, cl →∞ (which corresponds to
c 0k → ), the inverse localization length decays as
2
3
c
.
32l
σ
λ (27)
Equation (27) shows that, for increasing values of the cor-
relation length cl , the localization length increases with the
third power of the correlation length, as numerically ob-
served in Ref. 19.
It is interesting to compare the inverse localization
length (24) with the expression given by the Eq. (19).
When one considers the Anderson model with site energies
(3), for = 0E the IK-formula gives
[ ]
2 3
c
2
c c
( )sinh= .
8 cosh( ) 1 cosh( )
IK
k
k k
σ
λ
+
(28)
For c 0l → , Eq. (28) reduces to
2
= ,
8IK
σ
λ
which coincides with the prediction of the Thouless formu-
la and differs from the Derrida–Gardner result (26). On the
other hand, in the limit c 1l >> Eq. (19) reproduces the cor-
rect result (27). This implies that the band-center anomaly
is gradually suppressed as the correlation length increases.
This effect is confirmed by the numerical data, as
shown by Fig. 1. The data obtained for c = 0l (uncorrelated
disorder) shows that, at the band center, the Lyapunov ex-
ponent exhibits a pronounced dip (the anomaly) which,
however, is essentially suppressed already for c = 1l .
To understand the physical reason behind the suppres-
sion of the band-center anomaly, it is useful to consider
how the invariant measure ( )ρ θ changes as cl increases. In
the present case, at the band center the invariant measure
for the phase variable is given by Eq. (20) with the pa-
rameter α taking the value (25). For c = 0l the invariant
distribution takes the limit form
1( ) =
12 3 cos(4 )
2
ρ θ
+ θ
K
(29)
which coincides with the result derived in [12] for the case
of uncorrelated disorder. For c 1l >> , on the other hand, one
obtains
4
c
1 1( ) = 1 cos (4 ) .
2 64l
ρ θ − θ +
π
(30)
Equation (30) shows that the invariant distribution becomes
quickly uniform for increasing values of the correlation
length cl . This explains why the anomaly is suppressed and
the standard formula (19), derived under the assumption of a
flat distribution for the angle variable, becomes valid.
The band-center anomaly is not suppressed only if the
site energies have correlations of the form (5). As shown in
[18], a similar effect occurs for disorder with exponentially
decaying, positive correlations. One should also add that
exponentially decaying correlations with alternating sign
can have the opposite effect and enhance the band-center
anomaly [18].
We conclude our analysis of the Anderson model (1)–(3)
with a remark on the difference between a truly exponen-
tially correlated disorder with the correlator of the form
(6), and the Sales-de Moura disorder with the correlator
(5). The power spectrum corresponding to the exponential
correlator (6) is
c
e
c
sinh( )
( ) = ,
cosh( ) cos(2 )
k
K
k
µ
− µ
(31)
Fig. 3. (Color online) The inverse localization length 1
loc= L−λ versus the energy E for two different values of correlation length: (a)
c = 0l and (b) c = 1l . Red circles represent numerical data, while blue continuous curves correspond to Eq. (19). The intensity of the
disorder in numerical simulations was set to 2 = 0.01σ .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 351
I.F. Herrera-González, F.M. Izrailev, N.M. Makarov, and L. Tessieri
that leads to the inverse localization length,
2
c2
c
2 c
sinh( )( )
= 2coth( )
4 ( ) cosh( ) 1
k
k
k
ασ
λ − α −
E
K
(32)
with
2
c
1 1= 1 .
2 cosh( )k
α −
(33)
In the limit c 0l → , Eq. (32) tends to the anomalous form
(26). For c 1l >> , however, Eq. (32) becomes
2
c
.
16l
σ
λ (34)
Comparing the limit forms (27) and (34), it is easy to see
that the localization length increases linearly with the cor-
relation length cl when the disorder is exponentially corre-
lated in the strict sense, whereas it increases with the 3
cl
when the disorder has correlations of the form (5).
This behavior of the localization length is matched by
the corresponding behavior of the invariant measure in the
limit of large correlation length. When the power spectrum
of the disorder has the form (31), the invariant distribution
is given by Eq. (20) with the parameter α assuming the
value (33). For c = 0l , one recovers the distribution (29).
When c 1l >> , however, the invariant distribution assumes
the form
( )2
1 1( ) = 1 cos 4 .
2 16 cl
ρ θ − θ +
π
(35)
Comparing the asymptotic forms (30) and (35), one can
see that when the correlator has the form (5) the invariant
distribution becomes uniform much faster for increasing cl
than in the case for disorder with the exponential correla-
tions (6).
5. Summary
In this paper we demostrated how the approach, devel-
oped in [18] for the band-center in the Anderson model,
can be applied to the quite specific case of correlated dis-
order studied in Ref. 19. Our analytical results, obtained
for a weak disorder, allow one to explain the anomalous
relation 3
loc cL l∝ between the localization length and the
correlation length which was numerically observed in [19].
Our analysis shows that this dependence is not completely
due to the exponential decay of the correlations, as was
claimed, but has to be partly ascribed to the polynomial
prefactor in the binary correlator (5).
We showed that the band-center anomaly is suppressed
for increasing values of the correlation length cl when the
binary correlator has either the form (5) or (6). The locali-
zation length at the band center increases with cl in both
cases; however, the polynomial prefactor in the binary
correlator (5) results in a quantitative difference between
the two types of disorder. Specifically, it gives rise to a
faster suppression of the anomaly for increasing cl and
generates an unusual dependence 3
loc cL l∝ .
Acknowledgements
I. F. H.-G., F. M. I. and N.M.M. acknowledge support
from the CONACYT (México) under grant No. CB-2011-
01-166382. L. T. acknowledges the support of CIC-
UMSNH grant for the years 2016-2017.
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352 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2
1. Introduction
2. Definition of the model
3. The Hamiltonian map approach
4. The band center anomaly in the Sales-de Moura model
5. Summary
Acknowledgements
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