  The theory of orbital magnetization is reconsidered by defining additional
quantities that incorporate a non-Hermitian effect due to anomalous operators
that break the domain of definition of the Hermitian Hamiltonian. As a result,
boundary contributions to the observable are rigorously and analytically taken
into account. In this framework, we extend the standard velocity operator
definition in order to incorporate an anomaly of the position operator that is
inherent in band theory, which results in an explicit boundary velocity
contribution. Using the extended velocity, we define the electrons' intrinsic
orbital circulation and we argue that this is the main quantity that captures
the orbital magnetization phenomenon. As evidence of this assertion, we
demonstrate the explicit relation between the nth band electrons' collective
intrinsic circulation and the approximated, evaluated with respect to Wannier
states, local and itinerant circulation contributions that are frequently used
in the modern theory of orbital magnetization. A quantum mechanical formalism
for the orbital magnetization of extended and periodic topological solids
(insulators or metals) is redeveloped without any Wannier localization
approximation or heuristic extension [Caresoli, Thonhauser, Vanderbilt and
Resta, Phys. Rev. B 74, 024408 (2006)]. It is rigorously shown that, as a
result of the non-Hermitian effect, an emerging covariant derivative enters the
one-band (adiabatically deformed) approximation k-space expression for the
orbital magnetization. In the corresponding many-band (unrestricted) k-space
formula, the non-Hermitian effect contributes an additional boundary quantity
which is expected to give locally (in momentum space) giant contributions
whenever band crossings occur along with Hall voltage due to imbalance of
electron accumulation at the opposite boundaries of the material.
