Discrete-scale invariance (DSI) is a phenomenon featuring intriguing log-periodicity that can be rarely observed in quantum systems. Here, we report the log-periodic quantum oscillations in the longitudinal magnetoresistivity (ρ_xx) and the Hall traces (ρ_yx) of HfTe₅ crystals, which reveal the DSI in the transport-coefficients matrix. The oscillations in ρ_xx and ρ_yx show the consistent logB-periodicity with a phase shift. The finding of the logB oscillations in the Hall resistance supports the physical mechanism as a general quantum effect originating from the resonant scattering. Combined with theoretical simulations, we further clarify the origin of the log-periodic oscillations and the DSI in the topological materials. This work evidences the universality of the DSI in the Dirac materials and provides indispensable information for a full understanding of this novel phenomenon.

中文翻译：

---

拓扑材料 HfTe5 中的对数周期量子磁振荡和离散尺度不变性


离散尺度不变性（DSI）是一种具有有趣的对数周期性的现象，在量子系统中很少观察到。在这里，我们报告了 HfTe ₅晶体的纵向磁阻 ( ρ _xx ) 和霍尔迹线 ( ρ _yx ) 中的对数周期量子振荡，揭示了输运系数矩阵中的 DSI。 ρ _xx和ρ _yx中的振荡显示出具有相移的一致 log B周期性。霍尔电阻中 log B振荡的发现支持了作为源自共振散射的一般量子效应的物理机制。结合理论模拟，我们进一步阐明了拓扑材料中对数周期振荡和DSI的起源。这项工作证明了 DSI 在狄拉克材料中的普遍性，并为充分理解这一新现象提供了不可或缺的信息。