The modern semiclassical theory of a Bloch electron in a magnetic field now encompasses the orbital magnetic moment and the geometric phase. These two notions are encoded in the Bohr-Sommerfeld quantization condition as a phase (λ) that is subleading in powers of the field; λ is measurable in the phase offset of the de Haas–van Alphen oscillation, as well as of fixed-bias oscillations of the differential conductance in tunneling spectroscopy. In some solids and for certain field orientations, λ=π are robustly integer valued, owing to the symmetry of the extremal orbit; i.e., they are the topological invariants of magnetotransport. Our comprehensive symmetry analysis identifies solids in any (magnetic) space group for which λ is a topological invariant, as well as the symmetry-enforced degeneracy of Landau levels. The analysis is simplified by our formulation of ten (and only ten) symmetry classes for closed, Fermi-surface orbits. Case studies are discussed for graphene, transition metal dichalcogenides, 3D Weyl and Dirac metals, and crystalline and Z 2 topological insulators. In particular, we point out that a π phase offset in the fundamental oscillation should not be viewed as a smoking gun for a 3D Dirac metal.
Alexandradinata, A., Wang, C., Duan, W., & Glazman, L. (2018). Revealing the Topology of Fermi-Surface Wave Functions from Magnetic Quantum Oscillations. Physical Review X, 8(1). https://doi.org/10.1103/PhysRevX.8.011027