Topology is the study of how an object’s fundamental properties do not change despite being subjected to stretching, bending, and other forms of deformation. It is sometimes called “rubber-sheet geometry”.
Imagine a ball and a donut, each made of rubber. They cannot be stretched or bent to form the other object unless some sort of cutting, breaking, or punching of holes takes place. They are, therefore, topologically different.
However, to a topologist, a donut and a coffee cup look the “same” as both objects can be manipulated to form the other. The applications of topology can range from the study of cellular structures in biology, to data analysis in computer science, and quantum physics.
In 2016, the Nobel Prize in Physics was awarded to David Thouless, Duncan Haldane, and Michael Kosterlitz for their work on “Topological Phase Transitions” and “Topological Phases of Matter.” The trio discovered that topological properties do exist at the quantum level; previously, scientists were unsure if the study of topology could be applied to subatomic particles as physics tends to operate differently at the quantum level.
Topological phases of matter are related to the electronic properties of materials and the behaviour of the electrons in the material when a magnetic field is passed through them. Their work has shown that there are materials known as topological insulators that have perfect electron transport, yet are not affected by impurities in the material that might hinder that transport ability. The trio’s discovery has therefore opened new perspectives on exotic phases of topological materials like topological insulators, metals, and superconductors.
There are three paradigmatic topological phases of matter: the quantum Hall effect, quantum spin Hall insulators, and Chern insulators. The quantum Hall effect and the quantum spin Hall insulators have been thoroughly studied by scientists and realized in 3-dimensional (3D) space in experiments. Chern insulators, on the other hand, still lack a 3D realization despite plenty of theoretical work that predicts their existence.
However, that changed in 2022 when a team of scientists, Associate Professor Zhen Gao from the Southern University of Science and Technology (SUSTech) and his collaborators from Singapore and China, reported the first realization of a 3D Chern insulator found during their research, which was based on magnetically tunable photonic crystals.
Their work, entitled “Topological Chern vectors in three-dimensional photonic crystals,” has recently been published in Nature, a top research journal covering all fields of science and technology.
The team found that 3D Chern insulators have several unique characteristics. Firstly, they discovered that it is an intrinsic bulk topological invariant in 3D topological materials. In mathematics, when something is described as invariant, it means that its property or function remains unchanged even when a specific transformation is applied; bulk here refers to the bulk modulus, a formula that describes how resistant something is to compression.
Secondly, the team found that the topological surface states of 3D Chern insulators in the surface Brillouin zone form torus knots or links, which is topologically distinct from the surface states of other 3D topological materials. Knots here refer to mathematical knots, not the ones our wired earphones end up tangled in. Mathematical knots have no ends, so they cannot be untangled. Knot theory, which is the study of mathematical knots, deals with how knots can be deformed without one part cutting through another, and one of those unique knots is a torus knot. Think of a torus as a donut shape and a torus knot as a series of knots winding around and through that donut shape several times in various longitudinal directions.
Figure 1. A depiction of a torus knot that formed on the surface Brillouin zone of 3D Chern insulators
Thirdly, 3D Chern insulators can be driven into an ideal Weyl semimetal – materials whose conduction and valence bands intersect at certain points known as Weyl nodes – using magnetic fields. Prior to this, this topological phase transition was inaccessible to other platforms.
Topological physics and the future of electronics
With this realization of 3D Chern insulators, a new era begins in the study of topological physics. Scientists will be able to develop or discover new materials that are topological insulators. These topological insulators will allow for the designing of computers and other electronic or photonic devices that can use lower amounts of power more efficiently and still produce the same or even greater amounts of data output. The discovery of new topological insulators also means that scientists can make more significant progress in the production of quantum computers, where these exotic topological phases of matter ensure that data in quantum computers are more stable, allowing for more accurate computation.
Dr. Gui-Geng Liu, a Ph.D. student from Nanyang Technological University (NTU) and Assoc. Prof. Zhen Gao from SUSTech, are the co-first authors of this paper. Prof. Baile Zhang and Prof. Yidong Chong from NTU, Prof. Yihao Yang from Zhejiang University, and Prof. Peiheng Zhou from the University of Electronic Science and Technology of China (UESTC) are the co-corresponding authors. SUSTech is the co-first affiliation.
This research was supported by the National Natural Science Foundation of China (NSFC), Excellent Young Scientists Fund Program (Overseas) of China, Singapore Ministry of Education (MOE), and SUSTech.
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