The Mott-Hubbard Insulator: localization and topological quantum order

An insulating state of condensed matter is characterized by localization of the center of mass of the electrons. This criterion can be addressed in terms of the ground state on a torus with boundary conditions $\Psi_{K}(\{x_{1}+L,x_{2}, \ldots\}) = exp( i K L) \Psi_{K}(\{x_{1},x_{2}, \ldots\})$. As shown by Kohn[1], in an insulator the energy is insensitive to $K$ as $L \rightarrow \infty$, whereas in an ideal metal it increases as $K^{2}$. In addition, Souza, et al. derived expressions for the localization length in terms of the wavefunction as a function of $K$. The present work generalizes the arguments to provide a fundamental distinction between ``band'' and ``Mott-Hubbard'' insulators. The criteria involve only counting of electrons and experimentally measurable quantities independent of models, and they lead to the requirement that a Mott-Hubbard insulator with no broken local symmetry must have topological quantum order.\\[4pt] [1] W. Kohn, Phys. Rev. 133, A171 (1964)\\[0pt] [2] I. Souza, et al., Phys. Rev. B 62, 1666 (2000).