Macroscopic magnetization is a fundamental concept that all undergraduates learn about in elementary courses. In view of this, it is truly extraordinary that before 2005 there was no generally accepted formula for the macroscopic orbital magnetization in condensed matter. Orbital magnetization--as opposed to spin magnetization--occurs whenever time-reversal symmetry is broken in the spatial wavefunction. For instance, in a ferromagnet the spin-orbit interaction transmits the symmetry breaking from the spin degrees of freedom to the spatial (orbital) ones; the two contributions to the total magnetization can be resolved experimentally. Other examples include the induced magnetization in applied magnetic fields, or in any other time-reversal-symmetry breaking perturbations. Whenever the unperturbed system is nonmagnetic, the induced magnetization is 100% of the orbital kind.
Sweeping advances are occurring these days in the field of orbital
magnetization, and a "modern theory" is in development. The key formulas are resemblant of (but more complex than) the Berry-phase formulas of the modern theory of electric polarization, developed in the 1990s. So far, formulas for orbital magnetization have been established for: (1) crystalline solids, either metallic or
insulating, at the mean field level (HF or Kohn-Sham); (2) noncrystalline insulators at a the mean-field level (such as for Car-Parrinello simulations). Some progress has been achieved even in the case of a correlated wavefunction, but the ultimate theory has not yet been developed.
As for implementations, only model Hamiltonians have been addressed so far; a first-principle implementation is under way at the time this proposal is written. Concerning applications of the novel theory, a promising novel scheme for evaluating NMR shielding tensors has been proposed; its first-principle implementation is also under way.
Another open issue relates orbital magnetization (which is a ground state property) to magnetic circular dichroism, by means of magneto-optical sum rules widely used by X-ray spectroscopists at synchrotron facilities. Related sum rules have been used to measure local orbital moments even in antiferromagnets, where the macroscopic magnetization is zero. A precise microscopic definition of local orbital magnetization is still lacking (for both ferromagnets and antiferromagnets).