The possibility of driving an Anderson metal-insulator transition in the
presence of scale-free disorder by changing the correlation exponent is
numerically investigated. We calculate the localization length for
quasi-one-dimensional systems at fixed energy and fixed disorder strength using
a standard transfer matrix method. From a finite-size scaling analysis we
extract the critical correlation exponent and the critical exponent
characterizing the phase transition.Comment: 3 pages; 2 figure

Alexander Croy

F. A. B. F. de Moura

I. F. dos Santos

Michael Schreiber

R. A. Römer

S. Russ

W. Liu

English

arXiv

The possibility of driving an Anderson metal-insulator transition in the presence of scale-free disorder by changing the correlation exponent is numerically investigated. We calculate the localization length for quasi-one-dimensional systems at fixed energy and fixed disorder strength using a standard transfer matrix method. From a finite-size scaling analysis we extract the critical correlation exponent and the critical exponent characterizing the phase transition

Croy, Alexander

Schreiber, M.

Chalmers Research

Correlation-strength-driven Anderson metal-insulator transition

Chalmers Publication Library

PHYSICAL REVIEW B 85, 205147 (2012)
Correlation-strength-driven Anderson metal-insulator transition
Alexander Croy*
Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany and
Department of Applied Physics, Chalmers University of Technology, S-412 96 Go¨teborg, Sweden
Michael Schreiber
Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany
(Received 18 January 2012; revised manuscript received 15 March 2012; published 31 May 2012)
The possibility of driving an Anderson metal-insulator transition in the presence of scale-free disorder by
changing the correlation exponent is numerically investigated. We calculate the localization length for quasi-one-
dimensional systems at fixed energy and fixed disorder strength using a standard transfer matrix method. From
a finite-size scaling analysis we extract the critical correlation exponent and the critical exponent characterizing
the phase transition.
DOI: 10.1103/PhysRevB.85.205147 PACS number(s): 71.30.+h, 72.15.Rn, 71.23.An
I. INTRODUCTION
The Anderson model of localization1 has been the subject
of intense study over the past few decades. In particular, the
occurrence of a metal-insulator phase transition (MIT) in three
dimensions (3D) has attracted a lot of interest.2,3 Theoretical
studies of the MIT have focused mainly on situations with
uncorrelated disorder.3–6 Therefore, one of the open questions
in the field is the role of long-range correlated disorder in the
Anderson MIT.
For uncorrelated disorder, the Anderson transition can
be driven either by increasing the disorder strength or by
changing the Fermi energy.3 In the former case, for sufficiently
strong disorder strength W > Wc(E) all electronic states are
exponentially localized, where the value Wc(E) depends on the
Fermi energy E. On the other hand, at fixed disorder strength,
states with |E| < Ec(W ) are extended and otherwise localized.
The presence of correlations provides an additional possi-
bility of achieving the MIT. Depending on the nature of the
correlations, a transition may, in principle, be also driven by a
change of the correlation strength or correlation length. Such
a scenario might be relevant in situations where the disorder
is induced by a complex environment surrounding the system
of interest.
In this article, we study the possibility of a correlation-
strength-driven Anderson MIT in 3D. We consider the case
of scale-free disorder, which is characterized by a power law
with correlation exponent α. We find at fixed energy and fixed
disorder strength that the localization length behaves as
λ(α) ∝ |αc − α|−ν , (1)
where the critical exponent ν depends on the values of W and
E. The obtained critical valuesαc are consistent with results for
disorder- and energy-driven MITs in the presence of scale-free
disorder.7
II. MODEL AND METHOD
To study the influence of scale-free disorder on the
Anderson MIT, we use the usual tight-binding Hamiltonian
in site representation1,3
H =
∑
i
εi|i〉〈i| −
∑
i j
tij |i〉〈j| , (2)
where |i〉 denotes a localized state at lattice site i. The hopping
matrix elements tij are restricted to nearest neighbors. As usual,
we set these elements to unity and thereby fix the unit of energy.
The onsite potentials εi are taken as random numbers with a
Gaussian probability distribution. Specifically, we use random
potentials with mean 〈εi〉 = 0 and a correlation function of the
form
C() ≡ 〈εiεi+〉 ∝ ||−α ( = 0) , (3)
where α is the correlation exponent. In the context of
Anderson localization, this correlation function has been
used to study localization in the presence of long-range
correlations for one-dimensional,8–16 two-dimensional,17–21
and three-dimensional7,22 systems.
For the numerical calculations we generate the onsite poten-
tials for systems of size M × M × L using a modified Fourier
filtering method (FFM).23 Additionally, we shift and scale
the resulting random numbers to have vanishing mean and
variance C(0) = W 2/12. We focus on quasi-one-dimensional
systems with L = 400 000 and M = 5,7,9,11, and 13. The
localization length λ is calculated using a standard transfer-
matrix method (TMM).3 Monitoring the variance of the change
of the Lyapunov exponent during the TMM iterations gives a
measure of the accuracy of the localization length.24 We use
a new seed for each parameter combination (E, W , α, M).
Lastly, the critical exponent and the critical correlation strength
are obtained from a finite-size scaling (FSS) analysis.25 We
expand the one-parameter scaling ansatz26 for the reduced
localization length  = λ/M
(M,τ ) = F (M1/νχ (τ ),M−yφ(τ )) (4)
into a Taylor series
(M,τ ) =
nI∑
n=0
φnM−nyFn(χM1/ν) , (5)
where χ is a relevant scaling variable, φ is an irrelevant
scaling variable, y > 0 is the irrelevant scaling exponent, and τ
205147-11098-0121/2012/85(20)/205147(3) ©2012 American Physical Society
ALEXANDER CROY AND MICHAEL SCHREIBER PHYSICAL REVIEW B 85, 205147 (2012)
-6 0 6
E
c
0
5
10
15
20
25
30
W
c
0.9
1.5
2.5
uncorrelated
FIG. 1. (Color online) Schematic phase diagram based on the
results reported in Ref. 7, which were obtained using standard TMM
calculations of the localization length of quasi-one-dimensional
systems at fixed energy for various values of the disorder and at fixed
disorder strength for various values of the energy. Subsequently, the
metal-insulator transition was determined by finite-size scaling as
described in the text. The phase-space points discussed in the text are
indicated by symbols (∗ and ×).
measures the distance from the critical point. However, instead
of using energy or disorder strength to measure this distance,
we utilize the correlation exponent, i.e., τ = |α − αc|/αc. The
functions Fn, χ , and φ are further expanded up to order nR,
mR, and mI, respectively. Taking nI > 0 allows us to consider
corrections to scaling due to the finite size of the sample, which
is reflected in a systematic shift of  with M in Eq. (5). Using
a least-squares fit of the expansion of the reduced localization
length  to the numerical data allows us to obtain the critical
parameters. We have chosen the number of parameters as small
as possible to get a reasonable fit. Although one does ad hoc
not expect the FSS analysis to be valid in the present case, we
find that it is working surprisingly well, as we will show in the
following.
III. RESULTS
We set E = const and W = const while varying α. The
chosen values of E and W are indicated in Fig. 1, which
shows a schematic phase diagram for the Anderson MIT in
the presence of scale-free disorder. The position of the point
(E,W ) rather close to one of the transition boundaries, which
are shown for α = 0.9,1.5,2.5 and for uncorrelated disorder,
provides a first estimate of the expected critical correlation
exponent αc.
In Fig. 2(a) the reduced localization length is shown in the
vicinity of the band center, E = 0, setting W = 26. From the
dependence on the system size M , a clear transition can be
seen. For small correlation exponents (α < 1.5) the reduced
localization length increases with increasing size M , while
for large exponents (α > 1.5) it decreases. In the former case,
this means that the localization length increases faster than the
system size upon increasing M so that in the limit M → ∞
0.5 1 1.5 2 2.5
α
0.4
0.5
0.6
0.7
Λ
5
7
9
11
13
(a)
2 2.4 2.8 3.2 3.6
α
0.46
0.48
0.5
0.52
0.54
0.56
0.58
Λ
5
7
9
11
13
(b)
FIG. 2. (Color online) Reduced localization length  vs correla-
tion exponent α. Solid lines show FSS fit to numerical data. (a) Taking
corrections to scaling into account (nR = 2, nI = 1, mR = 2, mI = 0)
for E = 0.0,W = 26.0. (b) Without taking corrections to scaling into
account (nR = 2, nI = 0, mR = 2, mI = 0) for E = 6.0,W = 16.5.
the wave functions will be extended; while in the latter case,
the localization length increases slower than the system size
which means that the wave functions remain localized in the
limit M → ∞, i.e., in the 3D case. The former behavior
is characteristic for a metallic phase and the latter for an
insulating phase. The FSS procedure yields for the critical
correlation strength αc = 1.44 ± 0.04, which agrees very well
with the value expected from the phase diagram. The actual
values of the expansion parameters, mR,I and nR,I, are given in
the figure caption. The critical exponent is ν = 0.98 ± 0.09
(y = 2.0 ± 1.3), which is different from the value ν0 =
1.58 ± 0.03 obtained for uncorrelated disorder25 and from
ν(α = 1.5) = 1.69 ± 0.22 reported for scale-free disorder,7
both taken at E = 0. These different critical exponents are
an indication that the behavior of ν is nonuniversal. Similar
observations have been made previously yielding different
critical exponents at the band center and near the band
edge.27,28
Also, for E = 6.0 and W = 16.5 we find a transition, as
shown in Fig. 2(b). In this case the critical value is found to
be αc = 2.85 ± 0.03, again consistent with the phase diagram
in Fig. 1. The critical exponent is ν = 0.62 ± 0.03, which is
even smaller than the exponent found at the band center.
205147-2
CORRELATION-STRENGTH-DRIVEN ANDERSON METAL- . . . PHYSICAL REVIEW B 85, 205147 (2012)
IV. DISCUSSION
Qualitatively, the correlation-strength-driven transition can
be understood by assuming an effective disorder strength
Weff(α), which depends on the correlation exponent. An
effective smoothing of the disorder potential has, for example,
been observed for 1D systems, where the localization length in
the band center increases for smaller correlation exponents.8,15
It is also in accordance with the shift of the phase boundary
towards higher energies and stronger disorder shown in
Fig. 1. Accordingly, the transition occurs when Weff(αc) =
Wc(E). Close to the transition, the localization length would
diverge according to λ ∝ |Weff(α) − Wc|−ν0 ∝ |α − αc|−ν0 ,
where we have expanded the effective disorder strength to first
order, Weff(α) ≈ Wc + (α − αc)∂Weff/∂α|αc . By construction
this procedure yields the correct critical correlation strength,
but it does not explain the deviation of the observed critical
exponents from the universal value ν0. We would like to note,
however, that a previous study of the MIT for uncorrelated
disorder already showed a dependence of ν on the location in
the phase diagram.27,28 One reason for the behavior observed
here could be a limited validity of the single-parameter scaling
ansatz. At least for 1D systems with correlated disorder it
was found that single-parameter scaling breaks down.16,29
However, we do not have any indications in our numerical data
about a breakdown of Eq. (4) in 3D, since the numerical scaling
works very well. Therefore, provided the one-parameter
scaling law holds in the presence of long-range correlations,
the discrepancy between ν and ν0 might also indicate that the
FSS method in the normally used form [i.e., based on Eq. (5)]
is not suitable for extracting the critical exponent in the present
case.
In summary, we have studied the influence of scale-free
disorder on the Anderson MIT at fixed energy and fixed
disorder strength. By varying the correlation exponent we
found an increasing reduced localization length for α < αc
and a decreasing reduced localization length for α > αc when
increasing the system size. A FSS analysis yielded critical
exponents which depend on the values of E and W and
are smaller than the universal value ν0 found previously for
uncorrelated disorder.25
*alexander.croy@chalmers.se
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205147-3


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Correlation-Strength Driven Anderson Metal-Insulator Transition

Correlation-Strength Driven Anderson Metal-Insulator Transition

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