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Title: Log-periodic quantum magneto-oscillations and discrete-scale invariance in topological material HfTe5
Authors: Wang, HC 
Liu, YZ
Liu, YJ
Xi, CY
Wang, JF
Liu, J
Wang, Y
Li, L
Lau, SP 
Tian, ML
Yan, JQ
Mandrus, D
Dai, JY 
Liu, HW
Xie, XC
Wang, J
Issue Date: 2019
Source: National science review, 6 Aug. 2019, v. 6, no. 5, p. 914-920
Abstract: 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 (rho(xx)) and the Hall traces (rho(yx)) of HfTe5 crystals, which reveal the DSI in the transport-coefficients matrix. The oscillations in rho(xx) and rho(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.
Keywords: Log-periodic oscillations
Discrete-scale invariance
Topological materials
Dirac materials
Hall resistance
Publisher: Oxford University Press
Journal: National science review 
ISSN: 2095-5138
EISSN: 2053-714X
DOI: 10.1093/nsr/nwz110
Rights: © The Author(s) 2019.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted reuse, distribution, and reproduction in any medium, provided the originalwork is properly cited.
The following publication Huichao Wang, Yanzhao Liu, Yongjie Liu, Chuanying Xi, Junfeng Wang, Jun Liu, Yong Wang, Liang Li, Shu Ping Lau, Mingliang Tian, Jiaqiang Yan, David Mandrus, Ji-Yan Dai, Haiwen Liu, Xincheng Xie, Jian Wang, Log-periodic quantum magneto-oscillations and discrete-scale invariance in topological material HfTe5, National Science Review, Volume 6, Issue 5, September 2019, Pages 914–920 is available at
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