Orbital Magnetization as a Local Property

The modern expressions for polarization $\P$ and orbital magnetization $\M$ are $\k$-space integrals. But a genuine bulk property should also be expressible in $\r$-space, as unambiguous function of the ground-state density matrix, "nearsighted" in insulators, independently of the boundary conditions---either periodic or open. While $\P$---owing to its "quantum" indeterminacy---is {\it not} a bulk property in this sense, $\M$ is. We provide its $\r$-space expression for any insulator, even with nonzero Chern invariant. Simulations on a model Hamiltonian validate our theory.