1 Discrete scale invariance of the quasi -bound states at atomic vacancies in a topological materia l

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Discrete scale invariance of the quasi-bound states at atomic

vacancies in a topological material

Zhibin Shaoa,1, Shaojian Lib,1, Yanzhao Liuc,1, Zi Lid,1, Huichao Wange, Qi

Bianb, Jiaqiang Yanf, David Mandrusf,g, Haiwen Liuh, Ping Zhangd,i,2, X.C.

Xiec,j,k,l, Jian Wangc,j,k,l,2 and Minghu Pana,b,2

aSchool of Physics and Information Technology, Shaanxi Normal University,

Xi’an 710119, China; bSchool of Physics, Huazhong University of Science and

Technology, Wuhan 430074, China; cInternational Center for Quantum

Materials, School of Physics, Peking University, Beijing 100871, China;

dInstitute of Applied Physics and Computational Mathematics, Beijing 100088,

China; eSchool of Physics, Sun Yat-sen University, Guangzhou 510275,

China; fMaterials Science and Technology Division, Oak Ridge National

Laboratory, Oak Ridge, TN 37831, USA; gDepartment of Materials Science

and Engineering, University of Tennessee, Knoxville, TN 37996, USA; hCenter

for Advanced Quantum Studies, Department of Physics, Beijing Normal

University, Beijing 100875, China; iSchool of Physics and Physical

Engineering, Qufu Normal University, Qufu 273165, China; jCollaborative

Innovation Center of Quantum Matter, Beijing 100871, China; kCAS Center for

Excellence in Topological Quantum Computation, University of Chinese

Academy of Sciences, Beijing 100190, China; lBeijing Academy of Quantum

Information Sciences, Beijing 100193, China.

Author contributions: J. Wang and M.H. Pan conceived and designed the

experiment. Z.B. Shao, S.J. Li and B. Qi performed the STM experiments.

M.H. Pan, J. Wang, Z.B. Shao, Y.Z. Liu, H.C. Wang analyzed the experimental

results. H. Liu, Y.Z. Liu, H.C. Wang, J. Wang and X.C. Xie contributed to the

theoretical discussion. Z. Li, and P. Zhang carried out the DFT calculations.

J.Q. Yan and D. Mandrus grew the samples. M.H. Pan and J. Wang wrote the

manuscript with inputs from all other authors.

The authors declare no conflict of interest.

1Z.S., S.L., Y.L. and Z.L. contributed equally to this work.

2To whom correspondence may be addressed. Email:

jianwangphysics@pku.edu.cn (Jian Wang), minghupan@snnu.edu.cn

(Minghu Pan), zhang_ping@iapcm.ac.cn (Ping Zhang)

Classification: Physical Sciences, Physics

Keywords: atomic collapse state | discrete scale invariance | topological

material | scanning tunneling microscope

Abstract

Recently, log-periodic quantum oscillations have been detected in

topological materials zirconium pentatelluride (ZrTe5) and hafnium

pentatelluride (HfTe5), displaying intriguing discrete scale invariance

(DSI) characteristic. In condensed materials, the DSI is considered to be

related to the quasi-bound states formed by massless Dirac fermions

with strong Coulomb attraction, offering a feasible platform to study the

long-pursued atomic-collapse phenomenon. Here, we demonstrate that

a variety of atomic vacancies in the topological material HfTe5 can host

the geometric quasi-bound states with DSI feature, resembling the

artificial supercritical atom collapse. The density of states of these

quasi-bound states are enhanced and the quasi-bound states are

spatially distributed in the “orbitals” surrounding the vacancy sites,

which are detected and visualized by low-temperature scanning

tunneling microscope/spectroscopy (STM/S). By applying the

perpendicular magnetic fields, the quasi-bound states at lower energies

become wider and eventually invisible, meanwhile the energies of quasi-

bound states move gradually towards the Fermi energy (EF). These

features are consistent with the theoretical prediction of a magnetic-

field-induced transition from supercritical to subcritical states. The

direct observation of geometric quasi-bound states sheds light on the

deep understanding of the DSI in quantum materials.

Significance Statement

The atomic collapse (AC) state in superheavy atoms is one of the most

fundamental problems remaining elusive in experiments. In materials with

relativistic Dirac carriers, quasi-bound states featuring discrete scale

invariance are proposed as the analogue of the AC state. Here, by using

scanning tunneling microscope/spectroscopy, we detect a series of

differential conductance peaks at geometric energies on atomic vacancies in

topological material hafnium pentatelluride (HfTe5), which represent discrete

scale invariant quasi-bound states. The geometric spatial distribution of these

states is also detected. Our experimental observations are consistent with the

AC model in Dirac materials. The direct imaging of geometric quasi-bound

states at the atomic scale sheds light on the deep understanding of the scale

anomaly and AC state in quantum materials.

Introduction

One of the most striking and challenging issues in fundamental physics is

the observation of supercriticality phenomenon in ultra-heavy nuclei which

does not exist in nature. The atomic wave function will collapse when the

nuclear charge parameter Z exceeds a critical value Zc (1-4). Unfortunately,

the very large Zc makes it almost impossible for atoms to satisfy the

supercritical condition that



=Z∙α exceeds a critical value of order unity,

where α∼1/137 is the fine structure constant. Recently, the Dirac materials

with massless or massive Dirac fermions analogous to high energy relativistic

particles provide new platforms for related investigations. Moreover, the

supercritical regime guarantees the existence of a geometric series of quasi-

bound states showing dramatic discrete scale invariance (DSI) property (5,6).

The DSI induces log-periodic corrections to scaling and has been rarely

demonstrated in quantum systems other than cold atom gas (7-14). In

condensed materials, the atomic collapse states were observed in graphene

by scanning tunneling microscope/spectroscopy (STM/S) experiments (15,16)

and further STM/S studies of graphene reported the two quasi-bound states

indicating a signature of DSI (17). In addition, the DSI feature has been clearly

detected in topological materials zirconium pentatelluride (ZrTe5) and hafnium

pentatelluride (HfTe5) by the observation of log-periodic quantum oscillation

involving up to five oscillating cycles in the magnetoresistance (MR) under

ultrahigh magnetic fields (B) (18-20). The underlying mechanism can be

attributed to the discrete scale invariant quasi-bound states composed of

relativistic quasiparticles and non-relativistic quasiparticles (or charge

impurity) in Dirac materials (18,21). Thus, Dirac materials can serve as

promising systems to uncover the DSI characteristic in quantum systems that

satisfy the supercritical collapse condition. Direct imaging of the quasi-bound

states responsible for the appearance of the DSI feature in topological

systems is certainly of particular importance while it has not been presented.

ZrTe5 and HfTe5 are predicted to be quantum spin Hall insulators in the two-

dimensional (2D) limit and the three-dimensional (3D) crystals are located

near the phase boundary between weak and strong topological insulators

(TIs) (22,23). However, previous experimental results about these materials

present a large diversity. Some angle-resolved photoemission spectroscopy

(ARPES) (24,25) and magnetoinfrared spectroscopy studies show that ZrTe5

is a Dirac semimetal (26). Negative magnetoresistance and the anomalous

Hall effect also support the massless Dirac band structure of ZrTe5 (27-29).

Recently, more and more electrical transport, ARPES and STM results

suggest that ZrTe5 and HfTe5 are TIs with a small gap (30-35). The small

Fermi surface and ultralow carrier density of ZrTe5 and HfTe5 guarantee the

supercritical Coulomb attraction due to the weak screening effect (18), which

makes them promising platforms to explore the quasi-bound states with DSI.

Results

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1Discretescaleinvarianceofthequasi-boundstatesatatomicvacanciesinatopologicalmaterialZhibinShaoa,1,ShaojianLib,1,YanzhaoLiuc,1,ZiLid,1,HuichaoWange,QiBianb,JiaqiangYanf,DavidMandrusf,g,HaiwenLiuh,PingZhangd,i,2,X.C.Xiec,j,k,l,JianWangc,j,k,l,2andMinghuPana,b,2aSchoolofPhysicsandInformationTechnology...

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