### Electrons in a Crystal Exhibit Linked and Knotted Quantum Twists

Physicists are uncovering an infinitesimally small universe made up of a bizarre and startling assortment of linkages, knots, and twisting as they go deeper into the quantum realm. Skyrmions are magnetic whirls that appear in some quantum materials and are frequently referred to as “subatomic hurricanes.” Others have twisted superconductivity that forms vortices.

A Princeton-led team of scientists has now revealed that electrons in quantum matter may attach to each other in odd new ways, according to a research published in the journal Nature. The research pulls together ideas from three different fields of study — condensed matter physics, topology, and knot theory – in a novel way, posing unanticipated concerns regarding electronic systems' quantum features.

The subject of theoretical mathematics known as topology explores geometric qualities that can be distorted but not altered inherently. Three scientists, including Duncan Haldane, the Thomas D. Jones Professor of Mathematical Physics and Sherman Fairchild University Professor of Physics at Princeton, were awarded the Nobel Prize in 2016 for their theoretical prediction of topology in electronic materials.

Since then, scholars have worked to broaden this field of study in order to get a better understanding of quantum mechanics, such as in the subject of "quantum topology," which aims to explain an electron's state as defined by its wave function. According to M. Zahid Hasan, the Eugene Higgins Professor of Physics at Princeton University and the study's principal author, this was the spark for the current research.

“We’re studying properties related to the shape of the wave functions of electrons,” said Hasan. “And we have now taken the field to a new frontier.”

A quantum mechanical structure known as a Weyl loop, which includes the wrapping of massless electron wave functions in a crystal, is a key component of this new frontier. The massless Weyl loops were identified in a molecule made up of cobalt, manganese, and gallium, with the chemical formula Co2MnGa, in a groundbreaking work reported in Science in 2019. Hasan directed this investigation, which includes several of the authors of the recent paper. They realized at the time that applied electric and magnetic fields cause strange behaviour in massless Weyl loops. These actions lasted until the room temperature was reached.

A Weyl loop is an example of a well-known type of quantum wave function looping in and of itself. “ Previous examples of topology in physics often involved the winding of quantum mechanical wave functions,” said Hasan, who led the current study. “These have been the focus of the physics community for at least the past decade.” These concepts are based on the team's previous work on rhodium and silicon (RhSi) crystals, as well as terbium, magnesium, and tin-based materials known as Chern magnets (TbMn6Sn6). Professor Hasan's team led both of these findings, which were published in Nature in 2019 and 2020, respectively.

The situation of Co2MnGa, on the other hand, found out to be distinct from the wave function winding discussed in traditional topological theories. “Here instead we have linked loops — our newly discovered knotted topology is of a different nature and gives rise to different mathematical linking numbers,” Tyler Cochran, a doctoral student at Princeton's Department of Physics and one of the study's co-authors, explained.

Professor Claudia Felser and her colleagues at the Max Planck Institute for Chemical Physics of Solids in Germany grew the Co2MnGa materials.

When the Princeton team calculated and realized that certain quantum materials, such as Co2MnGa, could house multiple Weyl loops at the same time, they had a breakthrough. “When multiple Weyl loops co-exist, it becomes natural to ask whether they can link up and knot in certain ways,” Hasan added.

Hasan's team's discovery sparked fundamental questions about linked Weyl loops, prompting the gathering of a team of experts in photoemission spectroscopy, mathematical topology, quantum material synthesis, and first-principles quantum calculations from around the world to better understand link topology and knotting in quantum matter.

### What’s knot to like

The worldwide team of academics worked together for more than five years to improve on their previous work on topological magnets in order to see the relationship empirically. The researchers used cutting-edge synchrotron radiation facilities in the United States, Switzerland, Japan, and Sweden to conduct sophisticated photoemission spectroscopy investigations.

“It turned out to be a fascinating puzzle that kept us hooked for a while,” said Ilya Belopolski, the study's main author and a postdoctoral researcher at the RIKEN Center for Emergent Matter Science near Tokyo, Japan. “Unraveling the intricacies of this elaborate linked quantum structure itself required more than three years of high-precision and ultra-high-resolution measurements at the world’s leading spectroscopic facilities.”

The testing results indicated a strange object that was folded in on itself and wrapped around a higher-dimensional torus. “Understanding the object’s structure required a new bridge between quantum mechanics, mathematical topology and knot theory,” said Guoqing Chang, a research author and assistant professor of physics at Nanyang Technological University in Singapore. Chang, a former postdoctoral researcher at Princeton who worked with Hasan, performed one of the first theoretical explorations of link topology in a groundbreaking paper published in Physical Review Letters in 2017.

In fact, the researchers discovered that current quantum materials theory was unable to appropriately explain the formation of this structure. Knot theory, on the other hand, they identified as a possible source of information.

“We came to realize that some aspects of knot theory are very powerful in explaining quantum properties of topological materials that were not understood before,” Hasan explained. “This is the first example that we know of where knot theory has been applied to understand the behavior of topological magnets. And this a very exciting!”

The discoveries add to the decades-long debate between physics and topology, this time introducing new mathematical theories to explain quantum ferromagnet investigations.

“Historically, some of the most important scientific discoveries arose when humans noticed new connections between mathematics and natural phenomena. It’s always exciting to find unexpected examples of subtle mathematics in our experiments", Hasan remarked. “Even more so, it was interesting that the mathematical connection was in the field of topology, which has continued to emerge time and again in different guises in the study of quantum materials.”

The researchers want to continue their study in a variety of avenues. Despite the fact that Hasan and his colleagues concentrated their work on topological magnet behavior, they believe the theory has the potential to explain additional quantum phenomena. “We believe that knot theory can also be applied to many other topological conductors, superconductors, qubits, and many other things,” he added.

Although the researchers were not considering practical applications — “We were involved in fundamental research,” Hasan said — their findings might aid in the development of quantum computing, particularly in the development of new types of topological qubits.