Orbital Magnetization from Uniform and Periodic Magnetic Fields

Magnetization is thermodynamically defined as the derivative of the grand potential with respect to a uniform magnetic field. However, a uniform magnetic field makes the kinetic momentum operators noncommuting and Landau-quantizes the electron motion. This changes the zero-field momentum-space to Landau-levels and raises a fundamental question: how can the thermodynamic response to a uniform field be reproduced by a linear-response calculation carried out in the momentum space of the zero-field problem? We address this question analytically in a quantum Hall ferromagnet that allows the orbital magnetization $M$ to be computed in a closed form. We first compute $M$ from the local Hartree--Fock projector response to a periodic magnetic field with zero net flux. We then compute $M$ from the derivative of the grand potential with respect to a uniform magnetic field along the St\v{r}eda line. The two approaches give the same result, even though the first keeps the Hilbert space fixed while the second changes the Landau-level degeneracy. Their agreement suggests that we should view orbital magnetization as the energy associated with the spectral flow that gives rise to the St\v{r}eda formula. Our work provides a tutorial introduction to orbital magnetization and its relation to the St\v{r}eda formula.