| dc.description.abstract | It is hard to conclude and give constructive critical remarks on a subject, especially at a stage when all the “easy” things have been done and the “hard” things left untouched. As mentioned in the Introduction, what is needed at this stage in this field is both providing a consistent picture and the presentation of a definite point of view. We feel, at least to some extent, that we have met the first requirement.
Now we turn to the second. As could be seen from the Introduction, the point of view is frankly biased. It was meant more to be an expression of our own ideas rather than a systematic presentation. In the same tone, we mention what should be, in our opinion, the future lines of pursuit. We will concentrate only on the localization aspect.
One of the foremost and controversial physical problems regarding the transport aspect—the abruptness or otherwise in the behaviour of the conductivity at the mobility edge—is closely related to the localization problem.
The problem, actually speaking, is to give Anderson’s approach a sound mathematical footing. The usual method of showing the existence and of the calculation of Ec, as was mentioned already, is not satisfactory from a mathematical point of view. Attempts should be made to study the random function (e.g., self-energy) in question itself, and not its study in terms of the divergence of a series. Apart from the self-consistent theoretical approach, an attempt was made in this direction by Papatriantafillou et al., where they studied the probability distribution of the propagator itself through an integral equation. But it was done only for one dimension, and an extension to higher dimensions should be sought.
Another approach, mathematically very elegant and directly tied to Anderson’s description of localized states (in terms of the peculiar behaviour of poles and residues), was put forward by Ishii and discussed later by Luck. Here the localized energy region of the spectrum of the Hamiltonian is characterized by a singularly continuous measure, i.e., the energy measure having its support in a kind of Cantor set. Ishii has shown that in one dimension, the whole spectrum becomes singularly continuous whatever the disorder is. This criterion was also used in the context of thermal conductivity of disordered harmonic chains by Casher and Lebowitz. Again, an extension of this approach to higher dimensions is needed. This will be a fruitful and aesthetically more pleasant approach.
One does not mind resorting to approximations in a basically exact approach, but one does not like to derive conclusions from a fundamentally non-rigorous study such as examining the behaviour of self-energy in terms of the divergence of a series. In a lighter vein, one is reminded of Abel, who opined: “Divergent series are an invention of the Devil. It is shameful to base on them any demonstration whatsoever.”
Finally, there is a dire need for exact results. Apart from the exact result in one dimension—that all states are localized for any disorder—there are no other results for the localization problem. Some kind of inequalities, like Bogolyubov’s inequality in phase transitions, are needed. It should be pointed out that this will not be a routine academic exercise. Though the localization–delocalization “transition” is in some sense similar to other types of order–disorder transitions known so far, in the sense that in both cases symmetry is broken, the symmetry breaking in the localizability problem is of an entirely different type. The transition is basically non-cooperative and non-ergodic in nature.
So one has to be very careful in defining terms like order parameter, transition, etc. This study may also improve our understanding of the phenomenon of glass, for according to a conjecture of Anderson, the same basically non-ergodic behaviour underlies this phenomenon too. | |
