Locality of the Density Matrix in Metals, Semiconductors, and Insulators

We present an analytical study of the spatial decay rate $\gamma$ of the one-particle density matrix $\rho (\overrightarrow{r},{\overrightarrow{r}}^{\prime })\sim \exp (-\gamma |\overrightarrow{r}-{\overrightarrow{r}}^{\prime }|)$ for systems described by single-particle orbitals in periodic potentials in arbitrary dimensions. This decay reflects electronic locality in condensed matter systems and is also crucial for $O\left(N\right)$ density functional methods. We find that $\gamma$ behaves contrary to the conventional wisdom that generically $\gamma \propto \surd \Delta$ in insulators and $\gamma \propto \surd T$ in metals, where $\Delta$ is the direct band gap and $T$ is the temperature. Rather, in semiconductors $\gamma \propto \Delta$, and in metals at low temperature $\gamma \propto T$.