Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-met allic systems. We further discuss the case of density matrices for met allic systems at positive electronic temperature. A few other possible applications are also discussed.
We study the behavior of several physical properties of the Haldane model as the system undergoes its transition from the normal-insulator to the Chern-insulator phase. We find that the density matrix has exponential decay in both insulating phases, while having a power-law decay, more characteristic of a met allic system, precisely at the phase boundary. The total spread of the maximally localized Wannier functions is found to diverge in the Chern-insulator phase. However, its gauge-invariant part, related to the localization length of Resta and Sorella, is finite in both insulating phases and diverges as the phase boundary is approached. We also clarify how the usual algorithms for constructing Wannier functions break down as one crosses into the Chern-insulator region of the phase diagram. The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under consideration. The same equivalence implies that Chern insulators cannot display exponentially localized Wannier functions. An explicit condition for the reality of the Wannier functions is identified.