Space and Physics

An Apparently Ordinary Crystal Contains Knotted Quantum Loops Barely Imagined Previously


Stephen Luntz

Freelance Writer

clockMay 24 2022, 11:56 UTC
quantum topology

Diagram of the quantum electronic link in momentum (velocity) space observed in the topological Weyl magnet Co2MnGa, determined from advanced photoemission spectroscopy measurements. Image Credit: Ilya Belopolski and M . Zahid Hasan, Princeton University

A crystal with a deceptively simple formula has been revealed as displaying unexpectedly complex quantum behaviors. The interactions between electrons within the crystal go far beyond anything previously seen or predicted. Explaining them requires delving into areas of mathematics previously thought to bear little to no relationship to subatomic particles.


According to the model of particles we learn at school, electrons should repel each other, since they both have negative charge. However, under certain circumstances they can bind together to form Cooper pairs, whose joint movement overcomes resistance, leading to the immensely important phenomenon of superconductivity

Physicists have started to discover electrons in crystals can link together in far more exotic ways as well. A paper in Nature has stepped things up a notch, reporting on quantum loops that form themselves into knots and other shapes described by obscure niches of topology. Whether it will lead to anything as world-changing as superconductivity remains to be seen, but it's brought together previously unconnected areas of knowledge to even come close to understanding what is going on.

One of the central discoveries of quantum theory is that subatomic particles such as electrons behave both as particles and waves. Descriptions of their wave-like behavior are provided by wave functions. Quantum topology explores the shape of these waves, including a previously theoretical structure known as a Weyl loop formed by electron wave functions in crystals.

Many exotic quantum behaviors are only seen at temperatures close to absolute zero – as was the case with superconductivity for decades. However, in 2019, Weyl loops were described at room temperature in Co2MnGa crystal magnets. Cobalt, manganese, and gallium may not be the most familiar materials around, but they're also not that rare. To find something so complex in crystals made of just these three, in such a simple chemical ratio, represented quite a surprise. “High temperature” superconductors, by comparison, have formulas like Hg12Tl3Ba30Ca30Cu45O127.


Yet it seems Co2MnGa has more tricks up its sleeve. Where other examples of quantum topology involved winding wave functions something different is going on in Co2MnGa.

“Here instead we have linked loops — our newly discovered knotted topology is of a different nature and gives rise to different mathematical linking numbers,” said Princeton University graduate student Tyler Cochran, an author on the Nature paper, in a statement. A linking number is the number of times one curve winds around another.

“When multiple Weyl loops co-exist, it becomes natural to ask whether they can link up and knot in certain ways,” said Princeton's Professor Zahid Hasan.


Professor Hasan brought together a team capable of answering those questions, combining skills in areas previously as unrelated as photoemission spectroscopy – using synchrotron radiation to see what the materials were actually doing – and knot theory to make sense of the shapes observed. Plenty of quantum mechanics expertise was required to bridge the two. They observed three intertwined loop links within a three-dimensional torus and report; “Each loop links each other loop twice.”

“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 added

Certain mathematicians have been reported to say they chose their areas of expertise precisely because there were no real-world implications. Some with such intent have chosen topology, only to be repeatedly foiled. There's at least a chance this will happen again, with the work finding some as yet unidentified application in quantum computing or telecommunications.

Space and Physics
  • mathematics,

  • quantum,

  • crystals