Orbital magnetization in a supercell framework: Single {\bf k}-point formula

The position operator ${\bf r}$ is ill-defined within periodic boundary conditions: owing to this, both the macroscopic (electric) polarization and the macroscopic orbital magnetization are nontrivial quantities. While the former has been successfully tamed since the early 1990s, the latter remained a long-standing unsolved problem. A successful formula within DFT for crystalline systems has been recently found.\footnote{ D. Ceresoli, T. Thonhauser, D. Vanderbilt, R. Resta, Phys. Rev. B {\bf 74}, 024408 (2006).} The formula is based on a Brillouin-zone integration, which is discretized on a reciprocal-space mesh for numerical implementation. We find here the single ${\bf k}$-point limit, useful for large enough supercells, and particularly in the framework of Car-Parrinello simulations for noncrystalline systems. We validate our formula on the test case of a crystalline system, where the supercell is chosen as a large multiple of the elementary cell. Rather counterintuitively, even the Chern number (in 2d) can be evaluated using a single ${\bf k}$-point diagonalization.