Quantum oscillation in Hopf-link semimetals

Since the discovery of the relation between the Chern number and quantum Hall effect, searching for observables of topological invariants has been an intriguing topic. Topological Hopf-link semimetals have attracted tremendous interest, in which the conduction and valence energy bands touch at linked nodal lines. However, it is challenging to identify this sophisticated topology. We propose to use the quantum oscillation in strong magnetic fields to probe the Hopf links. For a generic model of a Hopf-link semimetal that captures the linked-trivial phase transition, we figure out the phase shifts of oscillation for all Fermi pockets in all magnetic-field directions, by presenting self-consistent results from the Fermi-surface tomography, Landau fan diagram, and electrical resistivity. As the magnetic field is rotated, the phase shifts exhibit a unique pattern, which could help to identify Hopf links in real materials, such as those in ${\mathrm{Li}}_{2}\mathrm{NaN}$.