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
lattices
of solids
[And58,And61,And78].
(A more recent review may be found in [LR85].) Here, I
discuss only those aspects of the theory that are relevant for the
understanding of
*localization phenomena*
on such lattices
that can be taken as simple model systems
for
*disordered* solids.

The concept of *transfer matrices* [Pen94] is an essential
ingredient of
the theory. As an introductory example for the use of these matrices I
consider a simplified discussion of
a stationary quantum wave function within
the one-dimensional
potential

In the zero potential region
between the -st and -th site
the
quantum
wave function can be written in the
form

(5.8) |

In this way the wave function in any of the inter-site regions of the lattice, given by , can be constructed systematically by specifying a suitable ``initial condition'' and iterating the discrete ``dynamical system'' (5.17). The particular properties of enter into the matrix elements of via the reflection and transmission coefficients , , at the -th site:

(5.10) |

(5.11) |

For a *periodic* potential
with
the conditions for complete
transmission through a lattice site are identical for all sites and can
be satisfied by adjusting the energy of the
electron.
This is
a consequence
of BLOCH's theorem which asserts the existence of delocalized
motion [Mad78].

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
another
class of
systems,
giving rise to another type of transfer matrices.
Here, the role of the
time-independent SCHRÖDINGER equation
is taken by
the *discrete SCHRÖDINGER equation*
[And58,And78]:

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)
again
has been
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
discuss
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
these
, 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
of the
complete lattice can
be made,

is a special case of the more general translation operator of equation (4.1b): .

Inserting
the expansion
(5.22) into
the SCHRÖDINGER equation
(5.21),
and making use of the orthonormality relation

(5.15) |

(5.16) |

Essentially, the expectation values , taken with respect to
, depend
on the weight factors of
only,

Up to this point,
the dynamics on the lattice as given by equation
(5.20)
is completely deterministic, and its
parameters are all fixed by specifying
.
On the other hand, since
a priori
the eigenstates
are unknown
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
just
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
equation
(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
discussion.^{5.5}

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

specify the interaction 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 for which all weight factors are identical; but in the present context this example expresses an oversimplification that rules out ANDERSON localization and therefore is not considered in the following.

Depending on
, the model may be simplified even
further.
In order to study ANDERSON localization it suffices to consider the
special case
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
approximation [Fis93].
(Cf. the discussion of tight binding on page
.)
The resulting *tight binding equation* is

Again, this time as a result of the tight binding approximation
(5.30a),
the problem can be formulated using
a
transfer matrix approach
by setting

the tight binding equation (5.31) becomes

Every point
(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 ) is obtained by

In such a setting FURSTENBERG's theorem
[Fur63]
can be applied, which deals with the more general class of unimodular
random matrices. Obviously the are unimodular for all ,

(5.26) |

for the ``LIAPUNOV exponent of the transfer matrix'' [Fis93] exists and that is positive.

with a normalization constant , and with coefficients 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 (5.40). 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 .
The
definition of a
tight binding model thus has to be completed by specifying the
distribution function for the
values of
.
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
B.
(In [FMSS85,CKM87]
ANDERSON localization is even proved under assumptions which are
somewhat weaker than those of FURSTENBERG's theorem, as discussed in
appendix B.)
Several different choices for satisfying these
requirements
have been applied successfully;
choosing a
Lorentzian distribution function -- in which case the model is
often referred to as the ANDERSON-LLOYD model --,

i.e. it depends on the energy and the half width of the Lorentzian only. A proof of this statement is reviewed in [Haa01].

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 I have 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.
- ...
discussion.
^{5.5} - 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. - ...
positive.
^{5.6} -
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 mapping (5.4) in the form of equation (5.35) and thus establishing positivity of the larger LIAPUNOV exponent of the system [Haa01].