The electronic properties of extended bulk materials such as metals and superconductors have been studied extensively by physicists. However, as the field has progressed from simple elemental materials to more complex synthetic ones, computational chemistry has come to the fore in addressing these problems. The "exotic materials" include the organic superconductors such as Bechgaard salts, polyacetylene, the fullerenes, and the high-temperature superconductors based on copper oxides. In addition, many new amorphous materials are technologically interesting in their electronic properties. For instance, electronic conduction in polymers, such as polyvinylcarbazole, is essential to some xerographic processes. The chemical complexity of these systems p uts a premium on understanding the fundamental physics in new ways that do not usually rely on the simple symmetries present in elemental materials. All of these systems exhibit electronic phase transitions as the chemical composition or doping is changed: their electronic states change qualitatively. Sometimes this transition is from being an insulator to a metal, sometimes from a metal to a superconductor or to some complex magnetically ordered structure.
One of the simplest electronic phase transitions is the transition between extended and localized states of a single electron moving in a random potential. Even though this problem is at the heart of the study of the electronic properties of any disord ered material, the traditional methods of simply combining rough semiquantitative theories and experiment have been insufficiently powerful to resolve all of the important issues. One reason is that real materials have many physical influences in addition to disorder, such as interactions of vibrations and interactions between different electrons in the same material. Some mathematical work, such as rigorous theorems related to one-dimensional Anderson localization, has already helped in understanding this problem. On the other hand, loopholes in these theorems have been uncovered when the disorder is of a special, correlated ki nd. For instance, certain 1-D systems do not have only localized states as the simple theorems had indicated. These unusual sorts of extended states arise in systems with certain kinds of correlated disorder or with quasi-periodic Hamiltonians.
Interestingly, far from being a mathematical curiosity, these exceptions to the simple theorems about one-dimensional localization seem to be at the heart of understanding the behavior of materials such as polyacetylene. In two- or three-dimensional materials, experiment has amply demonstrated the existence of both extended and localized states. However, there are still numerous c ontroversies about the applicability of simple phase transition ideas to these electronic phase transitions. Are they described by the usual scaling phenomenology of ordinary thermal phase transitions?
It has been argued that, in fact, such descriptions may break down because the wavefunctions at the transition are multifractal. Thus, the study of these electronic phase transitions has much in common with problems addressed in quantum chaos, where th e structure of the wavefunctions needs to be understood in a statistical way. The interacting electron systems and their phase transitions also carry mathematical challenges. In one dimension, the interactions between electrons again can cause them to b ehave as if they were insulating. These one-dimensional many-electron problems lead to exactly solvable Schrödinger equatio ns. The exact integrability of these classical models is connected with conformal invariance and the existence of solitons of nonlinear partial differential equations in one dimension. The question of whether such electronic interaction effects give rise to new phases for higher-dimensional systems doubtless has connections with the problem of solitons and exact integrable systems in higher dimensions--a problem that has attracted many in the area of partial differential equations.
Finally, a large number of interesting phase transitions occur in disordered systems that also have interactions. These include the Kondo effect in which isolated electronic sites behave as if they have spins that can be quenched, as well as the exotic spin glass phases that have proved useful at least as analogies in many other areas of chemistry and biology. At the moment, the most useful theory of these system is based largely on the use of the unrestricted Hartree-Fock approximation. There are numerous mathematical questions connected with the Hartree-Fock problem for such disordered systems. Many of the ideas invented by Lieb in his proof of the stability of matter may have practical use here. Again, the rigorous mathematical analysis can be of s ignificant help in understanding whether or not certain approximations can be used confidently in elucidating the qualitative physics. In addition, the same issues will arise when quantitative calculations are contemplated for specific materials.
The study of electronic phase transitions in clusters will make these issues even more pronounced. The usual theory of phase transitions concentrates on infinite systems, and only dominant thermodynamic contributions are computed. For clusters, the fin ite size effects and the asymptotics of the approach to the infinite limit become important mathematical problems.
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