Wang Jian of the School of Physics and his collaborators observed quasi-bound states with discrete scale invariance at atomic defects of topological materials

Professor Wang Jian and Academician Xie Xincheng of the Center for Quantum Materials Science, School of Physics, Peking University, Professor Pan Minghu, School of Physics and Information Technology, Shaanxi Normal University, Zhang Ping and Li Zi, Associate Researcher Li Zi of the Beijing Institute of Applied Physics and Computational Mathematics, Associate Professor Wang Huichao of the School of Physics, Sun Yat-sen University, and Professor Liu Haiwen of the Department of Physics of Beijing Normal University directly observed the quasi-bound state
with discrete scale invariance at the atomic defects of the topological material HfTe₅ 。 The research results were published online in the academic journal Proceedings of the National Academy of Sciences (PNAS
) on October 10, 2022, entitled "Discrete scale invariance of the quasi-bound states at atomic vacancies in a topological material.
"

The collapse of supercritical atoms produced by the interaction of superheavy atoms under strong coulombic interaction is an important prediction
of relativistic quantum mechanics.
Since superweight atoms that meet the supercritical conditions (Zα > 1, Z is the atomic number, and α ~ 1/137 is the fine structure constant) have not been found in nature, this phenomenon has not been directly confirmed
by experiments.
In 2018, Wang Jian's research team discovered a new log-periodic quantum oscillation (Science Advances 4, eaau5096 (2018)) in the magnetoresistance of the topological material zirconium telluride (ZrTe₅), and theoretically collaborated with Xie Xincheng's research group to reveal that the relativistic Dirac fermions and anisotropic charges form quasi-bound states under the interaction of strong coulombs, similar to unstable artificial superheavy atoms
.
This discovery indicates that topological material systems can be used as a new research platform
for studying the collapse phenomenon of supercritical atoms.

At the same time, the discovery of log-periodic quantum oscillations also reflects that supercritical quasi-bound states in Dirac materials have a novel physical property - discrete scale invariance or discrete scale symmetry.

Discrete scale symmetry is the breaking of continuous scale symmetry, accompanied by the characteristic feature
of logarithmic period.
In physics , discrete scale invariance can exist in self-similar fractal structures
in classical systems.
In quantum systems, discrete scale invariance requires the Hamiltonian of the system to meet the harsh conditions
of scale invariance and quantization at the same time.
Breaking from continuous scale symmetry to discrete scale symmetry is an example of quantum phase transition, which plays an important role in some fundamental quantum systems, such as the Schrödinger equation system with an inverse square (1/r²) potential, the Efimov effect
in the (2+1) dimensional quantum electrodynamic system and the cold atom system.
Therefore, the study of discrete scale invariance is of great significance
for understanding the deep physical laws of nature.
However, the experimental exploration of discrete scale invariance in many-body systems such as condensed matter systems is extremely challenging, and the corresponding experimental evidence is very limited
.

Scanning tunneling microscopy experiments can directly detect the change of local state density near charged impurities, and quasi-bound states similar to atomic collapse states have been detected in graphene in previous experiments, but because only two quasi-bound states have been observed, the characteristics of discrete scale invariance still need further experimental evidence (Note: only two states are difficult to show that the logarithmic relationship is satisfied).

Since 2018, Wang Jian's research team and collaborators have successively worked on the topological materials ZrTe 5 and hafnium telluride (HfTe₅ ) and its thin-sliced devices observed log-periodic quantum oscillations (up to five cycles), giving conclusive evidence of discrete scale invariance in condensed matter systems from the perspective of quantum transport experiments (Science Advances 4, eaau5096 (2018); National Science Review, 6, 914 (2019)； npj Quantum Materials 5, 88 (2020)）
。 On this basis, whether quasi-bound states with discrete scale invariance can be directly observed in topological materials is of great value
for the study of cutting-edge scientific problems such as relativistic-derived quantum states in Dirac quantum materials and supercritical atomic collapse with invariant discrete scales.

Wang Jian's research team and collaborators carried out systematic ultra-high vacuum scanning tunneling microscopy on
the topological material HfTe₅.
On the cleavage surface of the material, several types of charged impurity defects have been observed (Figures A and C).

At charged impurities that satisfy the supercritical condition, they observed a series of state density formants in the scanning tunneling spectrum (Panels B and D).

The number of formants in the scanning tunneling spectrum is as high as four, and the energy satisfies the isometric relationship (i.
e.
, logarithmic period), giving clear evidence of
quasi-bound states and discrete scale invariance at charged impurities.
After obtaining evidence of quasi-bound states and their discrete scale invariance at the atomic scale, the research team measured the spatial distribution of quasi-bound states (Figure E).

The spatial distribution radius of the quasi-bound state shows an equal proportional relationship consistent with the characteristic energy (Figure F), which further confirms the discrete scale invariance of the quasi-bound state
.
In addition, the team observed the response of the quasi-bound state to an applied magnetic field (Figure G).

With the increase of the external magnetic field, the formants corresponding to the quasi-bound state of lower energy gradually widen and eventually disappear, while the quasi-bound state gradually approaches the Fermi surface, which is consistent with the supercritical to subcritical phase transition phenomenon caused by the applied magnetic field predicted by theory (Figure H).

This work is the first direct observation of the discrete scale invariance of relativistic quasi-bound states (atomic collapse states) at the atomic scale, which opens up new ideas for the study of atomic collapse states, discrete scale invariances and novel quantum states in quantum materials, and is expected to stimulate more in-depth research and discussion
on discrete scale invariance in solid physical systems.

Fig.
: (a-d) High-resolution scanning tunneling microscope topography diagram (a,c) of atomic defects on the surface of topological material HfTe₅ and corresponding scanning tunneling spectrum (b,d): the number of formants in the scanning tunnel spectrum is as high as four, and the energy satisfies the proportional relationship (that is, logarithmic period); (e-f) The spatial distribution of state density near charged impurities (i.
e.
, atomic defects) at different energies (e) and the corresponding quasi-bound state radius (f) satisfying the invariance of discrete scales; (g) The evolution of the quasi-bound state energy at the charged impurity with the increase of the applied magnetic field: the formant corresponding to the lower energy quasi-bound state gradually widens and eventually disappears, while the quasi-bound state gradually approaches the Fermi surface; (h) Evolution of the theoretically calculated quasi-bound state energy (normalized using energy_E0, dimensionless) with magnetic field
.
The theoretical simulation results (h) are consistent with the experimental data (g).

Jian Wang, Minghu Pan and Ping Zhang are the co-corresponding authors of the work, and Dr.
Zhibin Shao from the School of Physics and Information Technology of Shaanxi Normal University, Dr.
Shaojian Li from Huazhong University of Science and Technology, and postdoctoral fellows Liu Yanzhao and Li Zi from the Center for Quantum Materials Science, School of Physics, Peking University, are the co-first authors
.
The main collaborators of this work include Xie Xincheng, Wang Huichao, Liu Haiwen, Professor Jiaqiang Yan and Professor David Mandrus of Oak Ridge National Laboratory in
the United States.
This work is supported
by the National Key R&D Program, the Postdoctoral Innovation Talent Support Program, the National Natural Science Foundation of China, the Beijing Municipal Natural Science Foundation, the Center for Excellence and Innovation of the Chinese Academy of Sciences, and the China Postdoctoral Science Foundation.