The elegant experiments on photonic localization in randomized photonic crystals (Physics Today, May 2007, page 22) raise an interesting question: What happens when an electronic system is strongly disordered but not random? Traditionally, “random” has been thought of as a convenient mathematical tool that gives a good approximation to any strongly disordered system, but a series of experimental studies, beginning in the 1980s with the metal–insulator transition in semiconductor impurity bands, have shown many properties that are inconsistent with random models. Thus the photonic experiments have confirmed idealized models of strong disorder and have simultaneously emphasized just how rare such idealized disorder really is.

Why shouldn't random models work just as well for electronic as for photonic systems? Electronic interactions are normally much stronger than photonic ones, but they often are also randomized by strong disorder. That is especially true in the presence of several kinds of incoherent disorder—for instance, mixed impurities, or impurity clustering, in the case of semiconductor impurity bands. For a long time it was thought that strong disorder is always incoherent, but modern materials science has become so sophisticated that it can produce not only rare examples of ideally randomized materials but also, and much more often, the opposite extreme: self-organized networks embedded in, and actually made possible by, strong disorder.

The classic examples of strongly disordered materials are found in network glasses, which William H. Zachariasen famously described in 1932 as “continuous random networks.” 1 If those networks were random, then increasing their connectivity—by replacing divalent atoms like calcium with tetravalent atoms like silicon—should lead to a stiffness transition, which was present in early data. Compositional proximity to a stiffness transition enhances the ability of the melt to develop longer-range order and avoid crystallization by nucleation of crystalline clusters. 2 More recently, purified samples have actually exhibited two stiffness transitions, usually spaced about 10% apart on a suitable connectivity scale. The two transitions bound a new kind of topological intermediate phase, not defined by any kind of conventional symmetry but only by its special non-random connectivity. The internal networks associated with this novel phase have properties of great technological importance: They are rigid but still free of internal stress. 3 That combination of properties makes it possible to produce window glass on large scales without having it crack or crystallize.

The discovery of self-organized networks in intermediate phases has generated “constraint theory,” a new kind of topological theory of strongly disordered solids. Constraint theory has a strongly non-Newtonian flavor. In crystals, many quantum and statistical properties are described by power-law scaling, but network self-organization is an exponentially complex combinatorial problem. (Mathematicians call such problems “nonpolynomial complete.” They are best addressed by the variational methods developed in the late 1700s by Leonhard Euler and Joseph Louis de Lagrange, methods that are the basis of Lagrangian mechanics.) In any case, the identification of many such systems has made possible the development of a new linear algebra for describing them; their properties are at best only marginally accessible to computer simulations on even the largest scales. 4  

The Physics Today story concludes on a positive note: our enhanced appetites for a deeper understanding of the transition between localized and delocalized states in materials. There are often two such transitions, and the intermediate phase between them is much more exciting than either transition alone.

1.
W. H.
Zachariasen
,
J. Am. Chem. Soc.
54
,
3841
(
1932
).
2.
J. C.
Phillips
,
J. Non-Cryst. Sol.
34
,
153
(
1979
).
3.
F.
Wang
,
S.
Mamedov
,
P.
Boolchand
,
B.
Goodman
,
M.
Chandrasekhar
,
Phys. Rev. B
71
,
174201
(
2005
).
4.
P.
Boolchand
,
G.
Lucovsky
,
J. C.
Phillips
,
M. F.
Thorpe
,
Phil. Mag.
85
,
3823
(
2005
).