Theory of the Insulating State

Abstract

In this paper a new and more comprehensive characterization of the insulating state of matter is developed. This characterization includes the conventional insulators with energy gap as well as systems discussed by Mott which, in band theory, would be metals. The essential property is this: Every low-lying wave function $\ensuremath{\Phi}$ of an insulating ring breaks up into a sum of functions, $\ensuremath{\Phi}={{\ensuremath{\Sigma}}_{\ensuremath{-}\ensuremath{\infty}}}^{\ensuremath{\infty}}{\ensuremath{\Phi}}_{M}$, which are localized in disconnected regions of the many-particle configuration space and have essentially vanishing overlap. This property is the analog of localization for a single particle and leads directly to the electrical properties characteristic of insulators. An Appendix deals with a soluble model exhibiting a transition between an insulating and a conducting state.