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ANDERSON Localization on One-dimensional Lattices
An important subject in solid state physics is the investigation of
electronic motion in disordered solids at low temperature.
ANDERSON initiated a particular type of research in this field,
focusing on one-dimensional lattices as model systems for the crystal
(A more recent review may be found in [LR85].) Here, I
discuss only those aspects of the theory that are relevant for the
on such lattices
that can be taken as simple model systems
The concept of transfer matrices [Pen94] is an essential
the theory. As an introductory example for the use of these matrices I
consider a simplified discussion of
a stationary quantum wave function within
consisting of a bi-infinite sequence of
at the equally spaced lattice sites
with the lattice constant .
For simplicity, the individual
assumed to be of the same
shape given by , but the weight factors
The single site potential is assumed to be sufficiently localized
such that the overlap of potentials belonging to
may be considered zero in
the inter-site regions.
In other words, an electron moving within such a lattice essentially
interacts at most with one
at any given time,
and it propagates freely
while not being scattered on-site.
A lattice of this type may serve as a model system for weakly bound
electrons moving within a disordered solid
in the framework of a single particle approximation; the lack of order
due to impurities
(e.g. different types of atoms [Fis96])
or other perturbations of the crystal lattice is
described by the varying , leading to reflection and transmission
from site to site.5.3See figure 5.5 for a schematic sketch of
In the zero potential region
between the -st and -th site
wave function can be written in the
the wave number
and some suitable
As the model does not depend on time explicitly it suffices to consider
The values of the coefficients in neighbouring flat regions
transfer matrix , which in this context is defined
In this way the wave function in any of the inter-site regions of the
can be constructed systematically by specifying
and iterating the
discrete ``dynamical system'' (5.17).
The particular properties of
enter into the
matrix elements of via the reflection and transmission
at the -th
The phases of these coefficients and the transmission probability
are subject of the shape of the -th
potential and depend on its width and depth, for example.
Due to the time-reversal symmetry of the system defined by the potential
(5.15), can be simplified to give
where the only remaining parameters are
and two phases
For a periodic potential
the conditions for complete
transmission through a lattice site are identical for all sites and can
be satisfied by adjusting the energy of the
of BLOCH's theorem which asserts the existence of delocalized
For aperiodic potentials, on the other hand,
which -- as discussed above -- provide a more appropriate model for
disordered solids than periodic potentials, the situation is
different and more conveniently discussed in terms of
giving rise to another type of transfer matrices.
Here, the role of the
time-independent SCHRÖDINGER equation
is taken by
the discrete SCHRÖDINGER equation
It describes a particle with energy
, moving on a
one-dimensional lattice of equidistant sites, and is to be solved for
, the probability amplitude of finding the particle at the
-th site. At each site a
acts as a diagonal energy;
the way in which this potential term
becomes ``randomized'' is discussed extensively below.
The off-diagonal matrix elements
of the sites labelled by and , respectively.
Often the are referred to as interaction energies
(cf. [Zim79]), although in general they cannot be restricted to
take on real values only.
The discrete SCHRÖDINGER equation (5.20) can
be derived from its continuous counterpart, the time-independent
SCHRÖDINGER equation for a state
with energy ,
and the electron mass
in the following way.
The potential (5.15)
assumed to be composed
of well localized, nonoverlapping individual potentials . This
assumption is often called the
tight binding approximation(see for example [Stö99])
-- although some authors also refer to tight binding
with respect to
a special case
of the model that I
below on page .
Let the eigenstates of be denoted by
with labelling the different eigenstates.
While a general state needs to be constructed by superposition of all of
, for the model system to be discussed here
it suffices to consider a single eigenstate,
taken to be normalized.
It may be looked upon as that single atom state interacting strongest
with the passing electron.
Obviously, the restriction to a single eigenstate is a further assumption
on the system, but allowed in the present context, since the resulting
model system is still powerful enough to explain some of the key features
of electronic states in disordered crystalline lattices.
The localization property of carries over to
such that the following
LCAO ansatz (linear combination of atomic orbitals, [Zim79])
for an eigenstate
complete lattice can
as a linear combination of the
shifted to each lattice site by means of the translation operator
is a special case of the more general translation operator
of equation (4.1b):
the SCHRÖDINGER equation
and making use of the orthonormality relation
which expresses the vanishing of the overlap between eigenstates belonging
to different sites,
I obtain the discrete SCHRÖDINGER equation
the matrix elements
and the diagonal energies
have been defined.
Essentially, the expectation values , taken with respect to
on the weight factors of
as the two expectation values
in equation (5.27),
taken with respect to
do not depend on
any more. This
observation is useful when discussing
the way in which disorder or ``randomness'' is introduced into the theory.
Up to this point,
the dynamics on the lattice as given by equation
is completely deterministic, and its
parameters are all fixed by specifying
On the other hand, since
and the calculation of the and might be
a difficult task, it is much simpler to choose these quantities in
an appropriate way, making the model as simple as possible, while still
retaining its essential characteristics.
In the following I
discuss the most important of the possible choices, which is known as
diagonal or site disorder.
Here, the matrix elements
are assumed to be ``constant'', i.e. they are not treated as
varying much with , . Frequently, most of the are taken
to be zero, and
a few of them take on a very limited number of
nontrivial values. A typical example for such a choice is given below in
equation (5.30). The disorder part of
the case of site disorder is constituted by assuming the diagonal energies to vary
with , in such a way that the are statistically distributed
according to a given distribution and nothing more is known about them.
It is a nice feature of this particular model that,
as indicated by
(5.27), the assumptions concerning the distribution
of the carry over to the , and vice versa. In this way
it is possible to adjust the disorder properties of the model by
specifying the weight factors of
the ``random potential''
in the very
beginning of the
Often, the model potential
is defined in such a
way that the resulting discrete SCHRÖDINGER equation is at least
approximately translation invariant
with the lattice constant ,
i.e. the matrix elements
depend on the distance of the two respective sites and only.
In these cases
the simplified discrete SCHRÖDINGER equation
can be used, where the
hopping matrix elements
of the particle at a given site with its -th nearest neighbour site.
An especially simple example
of such a model
is defined by a potential
all weight factors are identical;
but in the
an oversimplification that rules out ANDERSON localization and therefore is not considered in the following.
, the model may be simplified even
In order to study ANDERSON localization it suffices to consider the
where only the interaction with nearest neighbours is taken into account
and assumed to be symmetric,
The restriction (5.30a)
is also frequently referred to as the tight binding
(Cf. the discussion of tight binding on page
The resulting tight binding equation is
By scaling the energies and , can be set to unity
without loss of generality.
What is left is
a model system that contains just a single external parameter
, fluctuating with in a random fashion but in
accordance with some given distribution of values.
Again, this time as a result of the tight binding approximation
the problem can be formulated using
transfer matrix approach
Defining the transfer matrix as
tight binding equation (5.31)
(here: corresponding to the index )
to the right of some initial point
(with index )
on the lattice
can be reached by repeated application of the transfer matrix:
every point (with index )
to the left of another initial point
(with index )
In such a setting FURSTENBERG's theorem
can be applied, which deals with the more general class of unimodular
random matrices. Obviously the are unimodular for all ,
is a random variable for all
if and only if
The theorem then ensures that with probability one the limit
for the ``LIAPUNOV exponent of the transfer matrix''
[Fis93] exists and that is
positive.5.6(See appendix B for a more detailed exposition of
This means that in general
-- i.e. for generic values of the energy
essentially grow exponentially with .
For boundary conditions I choose the values of the components of
two vectors , describing distant points on
the lattice, i.e. with .
While by the above theorem both sequences
generically grow exponentially
-- thus yielding no physical (normalizable or at least improper)
it is still
possible to adjust the energy
in such a way that
the wave functions iterated from both ends match somewhere in between;
was first observed in [Bor63] and
rigorously proven in
to be the generic
As a result, one obtains almost surely a
discrete point spectrum
of energy eigenvalues with
eigenstates which are exponentially localized around
can be written as
with a normalization constant
, and with
the absolute values of which do not exceed unity
and typically oscillate rapidly with .
Note that in this way
the LIAPUNOV exponent
is identified as the inverse of the
localization length of the state described by
See figure 5.6 for a sketch of the envelope
of an ANDERSON-localized wave function.
A necessary condition for this explanation to work is that the diagonal
energies are randomly distributed under variation of .
definition of a
tight binding model thus has to be completed by specifying the
distribution function for the
Actually, in order to ensure that FURSTENBERG's theorem is
applicable, must be ``sufficiently well-behaved'' in a
way that is described in some more detail in appendix
ANDERSON localization is even proved under assumptions which are
somewhat weaker than those of FURSTENBERG's theorem, as discussed in
Several different choices for satisfying these
have been applied successfully;
Lorentzian distribution function -- in which case the model is
often referred to as the ANDERSON-LLOYD model --,
has the advantage that in this case an explicit expression for the
localization length can be derived analytically:
i.e. it depends on the energy and the half width of the
A proof of this statement is reviewed in
To summarize, a discrete SCHRÖDINGER equation describing electronic
motion within the framework of a one-dimensional lattice model of a
disordered solid has been derived, and
demonstrated that typical
solutions exhibit exponential localization.
- ... site.5.3
- The model can also be used to describe light
propagation in a randomly stratified transparent medium, where the medium
is found to reflect the light perfectly.
In this way an optical realization
of ANDERSON localization is obtained [BK97].
- ... lattice.5.4
- It is also possible to choose as a narrow
rectangular potential barrier, thus automatically
guaranteeing good localization of
[PGF85]. But this
choice requires a different physical interpretation
and is not used in the present context.
- Another way to introduce disorder into the
theory is to consider bond disorder. In this
case, the are assumed to be statistically
distributed, and the are constant.
Note that FURSTENBERG's theorem can also
be utilized to prove chaoticity of the classical kicked rotor
for sufficiently large , by reformulating the linearized standard
in the form of
equation (5.35) and thus
establishing positivity of the larger LIAPUNOV exponent of the system
Next: Mapping of the Quantum
Up: ANDERSON Localization in the
Previous: The Kicked Rotor
Martin Engel 2004-01-01