Evolution from Landau Quantization to Discrete Scale Invariance Revealed by Quantum Oscillations in Topological Materials
Dirac materials have been a unique solid state platform for exploring relativistic quantum phenomena including supercritical atomic collapse, which leads to emergent discrete scale symmetry and logperiodic quantum oscillations. In the relativistic regime, the fundamental effect in quantum electrodynamics, vacuum polarization, can further modulate the atomic collapselike state by screening bare charges but is rarely harnessed in condensed matter system. Here, we report a continuous progression from low field Shubnikov de Haas oscillations to high field log periodic oscillations in the Dirac material HfTe5, with both phenomena modulated by Fermi surface anisotropy. This maps the transition from single particle Landau levels to an interaction-driven, discrete scale invariant energy spectrum of quasi-bound states. Crucially, our findings suggest vacuum polarization provides a compelling mechanism for renormalizing the effective impurity charge, quantitatively explaining the carrier-density dependent scale factor. By revealing the intricate interplay between Landau quantization, many body electronic screening, and scale-symmetry breaking, our results establish Dirac solids as a controllable platform for exploring relativistic vacuum effects and emergent novel symmetry.