Topo-fermiology: revealing the topology of Fermi-surface wavefunctions from magnetic oscillations

In fermiology, the shape of the Fermi surface is deducible from the period of field-induced oscillations of the magnetization (dHvA) and resistivity (SdH). Here, we propose that the topology of the Fermi-surface wavefunctions are deducible from the phase offset $\lambda$ of these oscillations. $\lambda$ encodes not just the Berry phase, but also the orbital magnetic moment of wavepackets about their center of mass. In some metals and for certain field orientations, the symmetry of the orbit fixes $\lambda$ to integer multiples of $\pi$, i.e., $\lambda$ are the topological invariants of magnetotransport. Our comprehensive symmetry analysis identifies any metal for which $\lambda$ is a topological invariant, as well as identifies 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 dichalchogenides, crystalline/$\mathbb{Z}_2$ topological insulators, 3D Weyl/Dirac metals. We caution that the phase offset in the fundamental dHvA/SdH harmonic should not be viewed as a smoking gun for 3D Dirac metals. Alternative methods to extract $\lambda$ include the scanning-tunneling microscope and planar tunneling junction.