From electron cyclotron motion to discrete scale-invariant atomic collapse: bridging Shubnikov-de Haas and log-periodic quantum oscillations in topological HfTe₅/ZrTe₅

Log-periodic quantum oscillations (QOs) in condensed matter system reveal a unique discrete scale symmetry originating from supercritical atomic collapse quasi-bound states of relativistic Dirac fermions [1-4]. They typically emerge at low temperature and high magnetic field, conditions similar to those that produce the well-known Shubnikov–de Haas (SdH) effect, yet the relationship between the two phenomena has remained ambiguous, as they have not been observed together in a single system. Here, we resolve this standing question by elucidating the evolution of SdH and log-periodic QOs with varied hole concentration and magnetic field [5]. In particular, in relatively weak magnetic fields, we observe pronounced SdH QOs arising from cyclotron motion of carriers. Upon reaching the quantum limit, the same sample exhibit distinct log-periodic QOs. Notably, we find that the characteristic scale factor of the log-periodic QOs is tunable via carrier concentration, establishing a quantitative connection between carrier density and discrete scale invariance governed by vacuum polarization and Thomas–Fermi screening effects. By charting the transition from cyclotron motion to atomic-collapse-driven QOs, we establish a clear experimental bridge between these two regimes. Furthermore, magnetic torque measurements uncover signature of log-periodic QOs, highlighting the universal influence on magnetotransport. These results advance our understanding of scale symmetry broken quantum phenomena.