Rules for Phase Shifts of Quantum Oscillations in Topological Nodal-line Semimetals

Nodal-line semimetals are topological semimetals in which band touchings form nodal lines or rings. Around a nodal line, an electron can accumulate a nontrivial Pi Berry phase, so the phase shift in the Shubnikov-de Haas (SdH) oscillation may give a transport signature for the nodal-line semimetals. However, different experiments have reported contradictory phase shifts, in particular, in the WHM nodal-line semimetals (W=Zr/Hf, H=Si/Ge, M=S/Se/Te). For a generic model of nodel-line semimetal, we present a systematic calculation for the SdH oscillation of resistivity under a magnetic field normal to the nodal-line plane. From the analytical result of the resistivity, we extract general rules to determine the phase shifts for arbitrary cases and apply them to ZrSiS and Cu3PdN systems. Depending on the magnetic field directions, carrier types, and cross sections of Fermi surface, the phase shift shows rich results, quite different from those for normal electrons and Weyl fermions. Our results may help exploring transport signatures of topological nodal-line semimetals and can be generalized to other topological phases of matter.