Log-periodic quantum magneto-oscillations and discrete-scale invariance in topological material HfTe5

Abstract

Discrete scale invariance (DSI) is a phenomenon featuring intriguing log-periodicity which can be rarely observed in quantum systems. Here we report the log-periodic quantum oscillations in the magnetoresistance (MR) and the Hall traces of HfTe 5 crystals, which reveals the appearance of DSI. The oscillations show the same logB-periodicity in the behavior of MR and Hall, indicating an overall effect of the DSI on the transport properties. Moreover, the DSI feature in the Hall resistance signals its close relation to the carriers. Combined with theoretical simulations, we further clarify the origin of the log-periodic oscillations and the DSI in the topological materials. Our work evidences the universality of the DSI in the Dirac materials and paves way for the full understanding of the novel phenomenon.Discrete scale invariance (DSI) is a partial breaking of continuous scale invariance where observables of the system obey the scale invariance only for a geometrical set of choices written as the form of  n with  being the scaling ratio [1]. With the violation of the classical continuous scale symmetry, the DSI represents a scale anomaly, and the characteristic signature of DSI, the intriguing log-periodicity, exists in rupture, growth processes, turbulence, finance, and so on. The appearance of log-periodic structures indicates the characteristic length scales in a system, which is extremely interesting when it is fundamentally related to the underlying physical mechanism [1].The scale anomaly DSI is of high general interest while it can be rarely observed in quantum systems experimentally [2]. For a long time, the DSI has only been confirmed in cold atom systems and generated tremendous interest [3][4][5][6][7][8][9][10]. Nowadays, the DSI behavior in Dirac materials has also attracted attention in several subfields of physics [11][12][13][14][15][16]. Especially, the magneto-transport measurements on topological material ZrTe 5 reveal a new type of

Magnetotransport signatures of Weyl physics and discrete scale invariance in the elemental semiconductor tellurium

“…Log-Periodic MR/Hall Oscillations. Unusual properties of material systems under the quantum limit, wherein all of the carriers are condensed to the lowest Landau level, have been a topic of interest since their discovery (21,22,(39)(40)(41)(42)(43)(44). As a promising candidate for "Weyl semiconductor" with a relatively small carrier density and unique band topology, Te offers a suitable platform for the study of Weyl physics beyond the quantum limit.…”

“…4A shows the MR data taken under static fields at 1.8 K, which exhibit oscillations with more periods. Following previous studies (21,22), a suitable background (SI Appendix, Fig. S15A) obtained via smoothing was subtracted to better distinguish the oscillations (Materials and Methods).…”

“…For example, two well-known transport signatures in Dirac/Weyl semimetals are the chiral-anomaly induced negative longitudinal magnetoresistance (NLMR) (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and planar Hall effect (PHE) (12,(15)(16)(17)(18)(19)(20). Recently, a new type of quantum oscillations featuring log-periodicity, wherein the extrema magnetic fields of oscillations constitute a geometric series, was discovered in ZrTe 5 (21) and HfTe 5 (22) beyond the quantum limit. Physically, these unusual log-periodic oscillations can be attributed to the manifestation of discrete scale invariance (DSI) (21,23,24), whereby selfreproduction occurs at specific scales that form a geometric series.…”

Quantum transport in topological semimetals under magnetic fields (II)

“…In the quantum limit, our theory have shown that up to two resistance dips may appear if the system is a 3D strong topological insulator [12]. Surprisingly, recent experiments report up to five oscillations in the quantum limit [258,259]. The oscillation as a function of the magnetic field follows a logithimic scale invariance law, much like those in the Efimov bound states of cold atoms.…”

“…Efimovlike bound states have been used to understand the unexpected oscillations [260,261]. A direct fitting of the resistance in the experiment have also been demonstrated [258,259,261]. Nevertheless, other mechanisms that may lead to the scale invariance oscillations will be topics of broad interest.…”