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Discovery of Log-Periodic Quantum Oscillation: A New Chapter of Quantum Oscillations

Scientists have discovered log-periodic quantum oscillations in a solid-state system for the first time. This discovery represents the third known class of quantum oscillations after a nearly 90 year history of quantum oscillationmeasurements.


Background: quantum oscillations

In 1930, Lev Shubnikov and W. J. de Haas first observed that the resistance of a single crystal of bismuth changes periodically with the reciprocal of the magnetic field (1/B). This phenomenon is now known as Shubnikov-de Haas (SdH) magnetoresistance oscillations. Since then, the quantum oscillation experiments have gradually evolved into a powerful experimental technique to reveal novel physical phenomena in solid state systems. Over the past 90 years, researchers have observed quantum oscillations in diverse materials, including metals, metallic compounds, semimetals, semiconductors (and even insulators), as well as in artificial mesoscopic microstructures. To date, all known quantum oscillations can be classified into two classes: the first type is quantum oscillations with a periodicity in 1/B. A typical example is the SdH oscillations in magnetoresistance arising from Landau quantization, which has become an important experimental method to map out the Fermi surface and related physical properties of materials (See Figure 1 (A and B)). The second type is periodic in B, originating from the quasi-particle quantum interference in mesoscopic systems, such as the Aharonov–Bohm (AB) and Altshuler-Aronov-Spivak (AAS) oscillations in ring or cylinder structures in magnetoresistance (See Figure 1 (C and D)). Historically, each type of quantum oscillation reflects an important underlying physical mechanism and corresponds to new physical laws.


Discovery of Log-Periodic Quantum Oscillation

Recently, a research collaboration led by Prof. Jian Wang and Prof. Xincheng Xie at Peking University has discovered a new log-periodic quantum oscillation in high-quality single-crystal ZrTe5, a three-dimensional layered topological material. Under a strong magnetic field, transport carriers of the topological material enter the lowest Landau level, the so-called quantum limit. Beyond the quantum limit of ZrTe5, the research team discovered log-periodic oscillations involving up to five oscillating cycles (5 peaks and 5 dips) in magnetoresistance. This novel discovery is further confirmed by the magnetotransport results from different samples and different facilities with the maximum magnetic field up to 58 Tesla (See Figure 1 (E)).

The observed oscillations with the log-periodicity are different from all previously known quantum oscillations. Detailed analyses show that the log-periodic oscillations cannot be attributed to conventional type of quantum oscillations, even considering Zeeman splitting in the SdH mechanism. This indicates that the novel log-periodic quantum oscillation is a new member of the quantum oscillation family. Furthermore, the observed oscillations cannot be explained by previously known mechanisms beyond the quantum limit, such as the fractional quantum Hall states, Wigner crystal or density wave transition. This discovery potentially points to a new quantum state beyond the quantum limit.


Log-periodic quantum oscillation: discrete scale invariance in a quantum material

Further analyses reveal that the novel log-periodicity of the oscillating structures in the magnetoresistance demonstrates the existence of discrete scale invariance (DSI). A system with (continuous) scale invariance means that it can reproduce itself on different temporal or spatial scales. DSI is a partial breaking of continuous scale invariance, and subsequent observables of the system obey the scale invariance only for a geometrical set of characteristic length scales. The signature of DSI, log-periodicity, exists in animals, growth processes, financial crisis, earthquakes, turbulence, and so on. In a classical physical system, the DSI exists in the fractal structure induced by non-linear equations. Figure 2A shows a woodcut “Circle Limit III” by M. C. Escher, which represents a famous fractal structure ——the Poincaré disk model. In quantum systems, the best-known system with DSI is the Efimov trimer bound state, which has been observed in cold atom experiments and has inspired a growing research interest in related fields.


Log-periodic quantum oscillation: relativistic quantum quasi-bound states of supercritical atomic collapse

For solid-state systems, Dirac semimetals provide a promising platform to search for this rare and important phenomenon in quantum physics. On the other hand, relativistic quantum mechanics predicts that when the charge of a superheavy atomic nucleus Z exceeds a certain value satisfying Zα>1 (here α is the fine structure constant and α~1/137), the resulting strong Coulomb attraction causes an unusual atomic collapse state. This supercritical collapse state is a fundamental problem of nuclear physics, chosen by Science magazine as one of the 125 important scientific questions for humanity. However, this important theoretical prediction has not been directly confirmed in experiments. In Dirac materials, the quasi-particles (transport carriers) obey the relativistic equation, and the value of effective fine-structure constant in Dirac materials is much larger than the value in vacuum. Consequently, the supercritical collapse condition can be easily attained in Dirac materials, which provides a potential platform to study this peculiar phenomenon. In the Dirac material ZrTe5, the light holes obey the massless Dirac equation and their Fermi velocity is 450 km/s, much smaller than the speed of light in vacuum. Thus, the effective fine-structure constant of the system α>1, guaranteeing the supercritical collapse condition and giving rise to quasi-bound states with a finite lifetime. Meanwhile, these two-body quasi-bound states obey the DSI. These quasi-bound states in the supercritical regime can be considered as artificial atoms. Without magnetic field, these quasi-bound states obey DSI in both energy spectrum and radius. Under magnetic field, the energy of quasi-bound states approaches the Fermi surface at the corresponding magnetic field value obeying the geometric progression (See Figure 1F). The elastic scattering between the mobile carriers and the quasi-bound states around the Fermi energy strongly influences the transport properties, which gives rise to a log-periodic correction to the magnetoresistance. Furthermore, the SdH oscillations no longer survive beyond the quantum limit, and thus the signal of the log-periodic quantum oscillations can be clarified. Therefore, the discovery of the exotic log B periodic oscillations could virtually represent DSI for the quasi-particles in topological materials. Figure 2B shows the Russian doll analogy of the DSI in a Dirac system. Moreover, the investigation of these peculiar quasi-bound states in Dirac materials can broaden our understanding of supercritical atomic collapse.


Log-periodic quantum oscillation: a universal behavior in Dirac or topological materials

Based on this work, the research collaboration led by Prof. Jian Wang has found that the novel log-periodic oscillations and DSI features are potentially universal in topological materials with Coulomb attraction (arXiv:1810.03109). Hopefully, topological materials will become an interdisciplinary research platform to bridge the gap between the DSI and supercritical atomic collapse phenomenon. Thus, the discovery of the log-periodic oscillations provides a new experimental tool for exploring novel relativistic quantum phenomenon beyond condensed matter physics.


Publication and Acknowledgements

  The paper was published in Science Advances on November 2, 2018, Eastern Time. (Science Advances, 4,eaau5096 (2018).DOI: 10.1126/sciadv.aau5096) (Link: http://advances.sciencemag.org/content/4/11/eaau5096.)

Dr. Huichao Wang at Peking University and Prof. Haiwen Liu at Beijing Normal University contributed equally to this work. Prof. Jian Wang, Prof. X. C. Xie at Peking University are corresponding authors of this paper. Prof. Junfeng Wang and Prof. Liang Li at Wuhan National High Magnetic Field Center, Prof. Jiaqiang Yan and Prof. David Mandrus at Oak Ridge National Laboratory, Prof. Yong Wang at Zhejiang University and Prof. Jiyan Dai at The Hong Kong Polytechnic University collaborated on this project. We gratefully thank Prof. Robert Joynt at University of Wisconsin, Prof. Ziqiang Wang at Boston College, and Prof. Hui Zhai at Tsinghua University for helpful suggestions to the theoretical analysis.

This work was supported by the National Basic Research Program of China, the National Natural Science Foundation of China, the Research Fund for the Doctoral Program of Higher Education of China, the Open Project Program of the Pulsed High Magnetic Field Facility, the Key Research Program of the Chinese Academy of Sciences, the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, the Gordon and Betty Moore Foundation’s EPiQS Initiative and the Postdoctoral Fellowships Scheme of the Hong Kong Polytechnic etc.


Fig. 1. Quantum Oscillations Family. (A and B) Quantum oscillations with a periodicity in 1/B. A typical example is the Shubnikov-de Haas (SdH) oscillations arising from the Landau quantization in magnetoresistance. (A) From [1]. (C and D) Quantum oscillations with a periodicity in B, such as the Aharonov-Bohm (AB) and Altshuler-Aronov-Spivak (AAS) oscillations arising from the quasi-particle quantum interference in magnetoresistance. (C) From [2]. (E and F) Quantum oscillations with a periodicity in logB. Massless Dirac fermions with supercritical Coulomb attraction can give rise to the two-body quasi-bound states with discrete scale invariance. The quasi-bound states approach the Fermi energy under the magnetic field one after the other, and induce the log-periodic quantum oscillations in magnetoresistance.



[1] L. Schubnikow and W. J. De Haas, Nature (London) 126,500 (1930).

[2] R. A. Webb, S. Washburn, C. Umbach, and R. A. Laibowitz, Phys. Rev. Lett. 54, 2696 (1985). Theoretically proposed in 1959 by Aharonov and Bohm.


Fig. 2. Systems with discrete scale invariance (DSI). (A) “Circle Limit III” by M. C. Escher, which represents Poincaré disk model, a famous fractal structure. (From [3]) (B) Two-body quasi-bound states of massless Dirac fermions with supercritical Coulomb attraction. Inset is the Russian doll analogy of the discrete scale invariance in a Dirac system, the log-periodic quantum oscillations.


 [3] M. C. Escher, Circle Limit III, (1959).




1930年,舒伯尼科夫(Lev Shubnikov)和德哈斯(W. J. de Haas)首次在铋单晶材料中观测到电阻随磁场的倒数呈周期性变化的现象,这一发现被称之为Shubnikov-de Haas (SdH) 振荡。由此开始,输运实验中的量子振荡开始被物理学界所关注,并逐渐成为揭示新奇物理现象的一个重要研究工具。近90年中,研究者在包括金属,金属间化合物,半金属,半导体以及绝缘体在内的诸多材料及介观微结构中都发现了量子振荡。按照随磁场变化的规律可以把量子振荡分为不同的类型。目前已知的量子振荡有两大类:第一类量子振荡表现为随磁场的倒数呈周期性变化,起源于体系形成朗道能级,代表为磁电阻中的SdH振荡。目前SdH振荡已经成为探测材料费米面信息和物性的重要实验手段。参见图1(A,B)。第二类量子振荡表现为随磁场呈周期性变化,起源于介观系统中的准粒子量子干涉,例如环状和柱状结构磁电阻中的Aharonov-Bohm (AB) 振荡,Altshular-Aronov-Spivak (AAS) 振荡。参见图1(C,D)。历史上,每一类量子振荡的发现,都源于重要的物理机制,进而揭示出新的物理规律。





进一步分析表明,这一新奇发现中磁电阻振荡的对数周期性实质上是离散标度不变性的明显特征。标度不变性指体系在任何尺度下都是自相似的,体系不存在特定的特征尺度。离散标度不变性是连续标度不变性破缺的结果,其显著特征是体系的特征尺度满足等比数列。对数周期振荡是离散标度不变性的典型特征,这一特征在动物学、金融危机、地震、湍流等多种研究领域中都有所体现。在经典物理体系里,离散标度不变性存在于非线性方程导致的分形结构中。譬如著名数学物理学家庞加莱提出的庞加莱圆盘模型就是一种满足自相似性的分形结构,参见荷兰著名画家埃舍尔的画作Circle Limit III (图2A)。对于量子体系,目前已知的只有Efimov三体束缚态表现出离散标度不变的行为。近年来,Efimov三体束缚态在冷原子实验中得到了观测,进而激发了相关领域极大的研究热情。


在凝聚态物理领域,狄拉克材料体系为研究离散标度不变性提供了新的平台。另一方面,相对论量子力学预言当原子序数Z 满足不等式Zα>1时(其中α~1/137是自然界中的精细结构常数), 超重原子在强库仑势作用下发生塌缩现象。超重原子中的超临界塌缩现象是核物理领域的一个非常重要的研究课题,并入选Science杂志评选出的125个前沿科学问题。然而这一重要理论预言仍未得到实验的直接证实。狄拉克材料体系中的准粒子满足相对论性方程, 并且体系的精细结构常数远大于真空中取值, 大大降低了产生超临界坍缩态的临界电荷值,因而可以用来探究原子超临界塌缩现象。在狄拉克材料ZrTe5中,轻空穴满足无质量狄拉克方程为相对论粒子且有效光速约为每秒450公里,远小于真空中的光速。因此体系的精细结构常数α>1,满足超临界塌缩条件,会形成有限寿命的准束缚态。轻空穴受到电子或带电杂质的库仑吸引而形成准束缚态,这些准束缚态满足离散标度不变性。由于满足超临界塌缩条件,这些准束缚态可被看作是不稳定的人造原子。当外加磁场时,随磁场不断增加准束缚态的能级逐个经过费米面,并对费米面处的载流子(也即决定材料导电特性等物性的电子空穴等准粒子)引起散射,参见图1(F)。这些准束缚态在磁场作用下经过费米面而对载流子引起的共振散射会导致电阻发生振荡。数值计算表明,这些满足离散标度不变性的准束缚态经过费米面对应的特定磁场值也满足等比数列。因此,准束缚态与载流子之间的共振散射导致磁电阻发生对数周期量子振荡。当外加磁场超过量子极限后,通常的SdH振荡不再出现,因而更有利于观测到这种新的对数周期振荡。俄罗斯套娃可以看成是这种狄拉克体系中的离散标度不变性的一个生动类比,参见图2(B)。同时,这一研究表明拓扑材料体系可用于研究超临界原子塌缩现象,该体系中的准束缚态及其离散标度不变性具有新的特征,拓宽了我们对超临界原子塌缩的认识。




该工作于美国东部时间2018年11月2日发表于著名学术期刊Science Advances上(Science Advances, 4, eaau5096 (2018).DOI: 10.1126/sciadv.aau5096)。论文链接是:http://advances.sciencemag.org/content/4/11/eaau5096.

北京大学王慧超博士、北京师范大学青年研究员刘海文为共同第一作者,王健教授和谢心澄院士为文章共同通讯作者。该工作的合作者包括,武汉强磁场中心王俊峰研究员、李亮教授,浙江大学王勇教授,香港理工大学戴吉岩教授,美国橡树岭国家实验室Jiaqiang Yan教授、David Mandrus教授等。同时感谢威斯康星大学Robert Joynt教授,波士顿大学汪自强教授,和清华大学翟荟教授在理论上提供的诸多建议。

该工作得到了国家重点研发计划项目、国家自然科学基金、高等学校博士学科点专项科研基金、华中科技大学脉冲强磁场开放项目、中国科学院先导培育项目、US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division、Gordon and Betty Moore Foundation’s EPiQS Initiative、Postdoctoral Fellowships Scheme of the Hong Kong Polytechnic University等经费的支持。


图1:量子振荡家族。(A,B)随磁场的倒数周期性变化的量子振荡。典型代表为磁电阻中的Shubnikov–de Haas(SdH) 振荡,起源于朗道能级的量子化。(C,D) 随磁场周期性变化的量子振荡。代表为磁电阻中的Aharonov–Bohm(AB) 振荡和Altshuler-Aronov-Spivak(AAS) 振荡,起源于准粒子的量子干涉。(E,F)随磁场的对数周期性变化的量子振荡。物理机制为:无质量狄拉克费米子在超临界库仑吸引下形成两体准束缚态。这些满足离散标度不变性的准束缚态在磁场的影响下逐个经过费米面,引起磁电阻的对数周期量子振荡。


图2:具有离散标度不变性的系统。(A)  埃舍尔(M. C. Escher)版画“Circle Limit III”。代表了一种著名的分形结构——庞加莱(Poincaré)圆盘模型。(B) 无质量的狄拉克费米子受到相反电荷的重费米子的库仑吸引形成的两体准束缚态与对数周期量子振荡。插图为俄罗斯套娃,可以看作是狄拉克体系中的离散标度不变性的一个生动类比。