Surface sensitivity of magnetization in the mesoscopic regime

A solid's magnetization originates from unequal spin populations as well as orbital currents. In a localized representation some of orbital currents are flowing near the surface of the material, but their contribution to the magnetization scales with the size of the solid. Therefore, one may wonder if modifying the surface could change the solid's magnetization in proportion to the size of the entire solid. In this work we construct a simple model to study the orbital magnetic moment of a metallic solid. We generally find that modifying the surface of the solid causes two changes in the magnetization. The first is due to the change in the electron wavefunction. The second comes from the change in the kinetic angular momentum operator. When the model is in the macroscopic regime, the two effects mutually cancel and the magnetization is unaffected by the modifications on the solid's surface -- consistent with previous studies. However, in the mesoscopic regime, the two effects do not completely cancel each other out. In this regime, remarkably, when the surface of the metal is modified it induces an orbital magnetic moment whose magnitude scales in proportion to the size of the whole solid. In our model whenever Chern number is zero we include a correction term that reduces the norm of the commutator between the Hamiltonian and angular momentum operator.