  Valley magnetic moments play a crucial role in valleytronics in 2D hexagonal
materials. Traditionally, based on studies of quantum states in homogeneous
bulks, it is widely believed that only materials with broken structural
inversion symmetry can exhibit nonvanishing valley magnetic moments. Such
constraint excludes from relevant applications those with inversion symmetry,
as specifically exemplified by gapless monolayer graphene despite its
technological advantage in routine growth and production. This work revisits
valley-derived magnetic moments in a broad context covering inhomogeneous
structures as well. It generalizes the notion of valley magnetic moment for a
state from an integrated total quantity to the local field called "local valley
magnetic moment" with space-varying distribution. In suitable
inversion-symmetric structures with inhomogeneity, e.g., zigzag nanoribbons of
gapless monolayer graphene, it is shown that the local moment of a state can be
nonvanishing with sizable magnitude, while the corresponding total moment is
subject to the broken symmetry constraint. Moreover, it is demonstrated that
such local moment can interact with space-dependent electric and magnetic
fields manifesting pronounced field effects and making possible a local valley
control with external fields. Overall, a path to "local valleytronics" is
illustrated which exploits local valley magnetic moments for device
applications, relaxes the broken symmetry constraint on materials, and expands
flexibility in the implementation of valleytronics.
