Localization in All but the Pathological Cases

In the previous subsection implicitly the ``classical'' approach to the modelling by means of transfer matrices was followed: only the first component of the vector equations (5.115, 5.116, 5.119) was constructed to be equivalent to the tight binding equation (5.101), while the second component was made to be a trivial identity by construction; in addition, only the three amplitudes $\overline{u}_{E,m+2}$, $\overline{u}_{E,m}$ and $\overline{u}_{E,m-2}$ were allowed to contribute for any given $m$. In the present subsection I use a more general starting point for deriving transfer matrices which are more suitable for proving localization than $\cal{T}_m^{(2)}$, $\cal{T}_m^{\rm (2,left_1)}$ and $\cal{T}_m^{\rm (2,left_2)}$.

Generalized or higher order transfer matrix models reformulating the tight binding equation (5.101) can be introduced by considering $N$-dimensional transfer matrices (rather than the only two-dimensional transfer matrices as used up to this point) with some suitable $N\geq 2$:

The importance of condition (iii) is easily underestimated. It avoids the construction of models that are only consistent with the tight binding equation, without being equivalent to it. Such a situation occurs, for example, if one specifies the parameters in such a way that the only nontrivial subequation of (5.122c) is a nontrivial linear combination of (5.101); then it is not guaranteed that the solutions $\overline{u}_{E,m}$ of such a model also satisfy the tight binding equation.

This scheme allows to construct a multitude of different transfer matrix models. Once a model has been defined in this way, the remaining conditions of FURSTENBERG's theorem need to be checked, thus possibly leading to a proof of localization, if the model has been constructed properly.

Following this scheme, one can discuss the transfer matrix model with $N=3$ and $M=2$, for example. I have tried to choose the 18 parameters of this model in such a way that the resulting expressions are as simple as possible. After some (lengthy) algebra, the transfer matrix


The system (5.123) is suited for propagation in positive $m$-direction. Backpropagation can be achieved by using

It is easy to see -- by explicitly writing down all three subequations -- that the first subequation of the equation of motion (5.123c) contains the tight binding equation three times (in the sense of (iv)), the second subequation is equivalent to the tight-binding equation (satisfying condition (iii)), and the third subequation is just a trivial identity.

The $\cal{T}_m^{\rm (3)}$ do not share the worst disadvantage of the $\cal{T}_m^{\rm (2)}$ discussed in the previous subsection. Most matrix elements take on the trivial values 0 or 1, which are uncritical with respect to the integrability condition (B.2); the only nontrivial matrix elements, $\big(\cal{T}_m^{\rm (3)}\big)_{11}=-\big(\cal{T}_{m+2}^{\rm (3,left)}\big)_{32}$ and $\big(\cal{T}_m^{\rm (3)}\big)_{12}=-\big(\cal{T}_{m+2}^{\rm (3,left)}\big)_{33}$, are characterized by Lorentzian distributions of values, as figure 5.23 shows.

In fact, for both values of $\hbar$ considered in this figure, the approximating Lorentzians are nearly identical to those in figures 5.21a and 5.22a, respectively, where they approximate the distributions $p\big(\big(\cal{T}_m^{(2)}\big)_{11}\big)$. This indicates that the quotient of two consecutive nearest neighbour matrix elements -- which marks the only difference between $\big( \cal{T}_m^{(2)} \big)_{11}$ and $\big(\cal{T}_m^{(3)}\big)_{1j}$ -- does not take on values large enough to make $p\big(\big(\cal{T}_m^{(2)}\big)_{11}\big)$ and $p\big(\big(\cal{T}_m^{(3)}\big)_{1j}\big)$ differ significantly, an observation that is supported by figure 5.24, which displays typical values of $W_{m,m+2}/W_{m,m-2}$ for several values of $V_0$ and $\hbar$.

Apparently the distribution of all nontrivial matrix elements of the $\cal{T}_m^{\rm (3)}$ is governed essentially by the Lorentzian distribution generated by the (quasi-) random number generator (5.81).

Summarizing, for the parameters considered explicitly, and most probably for all parameter combinations belonging to the nonpathological systems with $W_{m,m+2} \neq 0$ for all $m$, $\cal{T}_m^{\rm (3)}$ and $\cal{T}_m^{\rm (3,left)}$ satisfy the integrability condition (B.2) and therefore give rise to quantum localization of the $T$-nonresonant kicked harmonic oscillator; the localization mechanism is identified as that of classical ANDERSON localization.