  The x-ray magnetic circular dichroism (XMCD) sum rule yields an extremely
useful ground-state observable, which provides a quantitative measure of
spontaneous time-reversal symmetry breaking (T-breaking) in a given material. I
derive here its explicit expression within band-structure theory, in the
general case: trivial insulators, topological insulators, and metals. Orbital
magnetization provides a different measure of T-breaking in the electronic
ground state. The two observables belong to the class of "geometrical"
observables; both are local and admit a "density" in coordinate space. In both
of them one could include/exclude selected groups of bands, in order to acquire
element-specific information about the T-breaking material. Only in the case of
an isolated flat band the contributions to the two observables coincide.
Finally, I provide the corresponding geometrical formula-in a different Hilbert
space-for a many-body interacting system.
