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Prof. M. Zahid Hasan
The 2021 Mustafa Prize Laureate

Revolution in Physics

Welcome to the mad, mad, mad, mad world of topological quantum matter

For so many years, topology was considered to have no or little practical value. Even mathematicians did not apply it to any serious problems in a serious way for a long time. For most of us, topology is nothing more than mind-boggling games with exotic geometrical objects like the Möbius strip – a loop made by twisting a ribbon once and gluing the ends together – which has only one surface and one edge. If you cut the Möbius strip lengthwise, you will get one larger loop instead of two loops of the same size. In other words, if you start to walk on a Möbius strip, after a 360o turn, you will find yourself on the opposite side of the start point, so it takes another 360o turn to return to where you start.

Another thing with the fascinating world of topology is that everything could morph into another geometrical object. The fine details of structures don’t matter in this world: a coffee mug morphs into a doughnut, whereas a glass morphs into a ball. This play-doh way of looking at things often has little connection to our daily lives. However, since the last decades of the 20th-century, topology started to appear in surprising places, from digital photographs, bank transactions, and biology to physics. Significantly, the marriage of topology and physics in the 21st century has proven to be highly fruitful. “We are in the middle of a topological revolution in physics, for sure,” Mohammad Zahid Hasan, the Eugene Higgins Professor of Physics at Princeton University, says. Hasan and his team had a key role in the flourishing development of this field.

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Some of the most fundamental features of subatomic particles turned to be topological at their heart. Consider the electron spin, which could be pointed up or down. Counter-intuitively, a 360o rotation will not return the particle to its original state. In the strange world of quantum physics, an electron is not just a particle and can also be described as a wave function, and a 360o rotation just shifts the crests and troughs of the wave. Therefore bringing an electron back to its initial state needs another complete turn. Does it sound familiar? Yes, if you are thinking of a Möbius strip. In fact, this is not just an analogy; it seems each electron does contain a tiny Möbius strip.

In the 1980s, some theorists began to suspect that a surprising newly discovered phenomenon called the quantum Hall effect might have a topological root. According to this effect, the electrical resistance of a single-atom-thick layer of a crystal changes in discrete steps. More importantly, temperature change or the crystal impurities don’t affect the resistance. “Such robustness was unheard of, and it is one of the key attributes of topological states that physicists are now eager to exploit,” Hasan says.

Instead of the Möbius strip in the case of electron’s spin, physicists revealed the topology of the quantum hall effect as the surface of a doughnut. Until the mid-2000s, physicists considered the quantum hall effect and other topological effects peculiar because they have been seen only in the presence of intensive magnetic fields. However, they realized if an insulator is composed of heavy elements, it is theoretically possible to have its own magnetic field due to the internal interactions between electrons and atomic nuclei. Because of the specific topology of the quantum state, such material could have a dual behavior against electricity: being a conductor on its surface just like metal and an insulator within its interior just like a plastic. No known material had ever behaved this way.

Once physicists realized making such material in a lab is actually possible, a race sat off. However, it didn’t take long. By 2008, Hasan and his team at Princeton University make the first real topological insulator out of bismuth antimonide crystal. “That was the beginning of the fun; the challenge is to find new materials that do not exist in nature,” he says.

The discovery was quite shocking even to physicists. Topological states now seemed to open a mysterious gate to a vast array of possibilities for discovering unknown effects in nature. In the past decade, researchers have found how topology can provide a unique insight into the physics of new exotic materials with unusual features. Now due to the groundbreaking works of Hasan and other pioneers, topological physics is truly exploding. According to the American Academy of Arts and Sciences (AAAS), “his experiments have been seminal in giving rise to the field of Topological Quantum Matter with more than 50,000 citations, which is now growing vigorously at the nexus of condensed matter physics, materials engineering, nanoscience, device physics, chemistry, and relativistic quantum field theory.”

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Thanks to the unusual mathematics that governs their behavior, in topological material, electrons can form certain states in which they collectively behave as a single elementary particle. These “quasiparticle” states may have properties not present in any known particle or even mimic particles that are not discovered yet. The major excitement was in 2015 when Hasan experimentally discovered one of the most-wanted quasiparticles in a topological semimetal: Weyl fermion, a massless fermion conjectured initially in the 1920s by the mathematician Hermann Weyl.

According to the standard model of elementary particles, all known fermions, including quarks, electrons, and neutrinos, have some mass. However, Hasan calculated that topological effects inside crystals of tantalum arsenide should create massless quasiparticles that act like Weyl fermions. Weyl fermion discovery named Top Ten Breakthrough of 2015 by Physics World. Being massless means this quasiparticle can move through a material faster than ordinary electrical currents and find applications such as ultrafast transistors or new kinds of quantum electronics and lasers.

Hasan has also made essential contributions in topological phase transitions, topological magnets, topological superconductors, and Kagome materials. Kagome lattices are formed by a network of corner-sharing triangles. When electrons are placed on such lattices, they exhibit several strange phenomena. The most striking of them is that some electrons behave as if they are massless.

When kagome lattice materials are magnetized, these massless electrons behave as if they are in a topological insulator. This is what makes them very interesting. “We are exploring kagome lattice materials in search of new types of topological insulators, especially looking for the ones that may remain topological at room temperature,” Hasan says. “Superconductivity on kagome lattices may be topological, so such materials may provide a new platform for qubits (quantum computing).”

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Hasan believes his research field is primarily discovery-driven, not application-driven. “Once we discover something unexpected, we try to explore it for deeper understanding.” However, finding pathways to develop applications of topological materials is always a shorter-term goal. There are two primary directions in this case: One is to discover a topological magnet that may work at room temperature and develop it further to make a low dissipation device. The other direction is to discover a topological superconductor and optimize it for quantum braiding operation for creating a functional topological qubit that is naturally fault-tolerant.

Hasan and his team are currently working on both directions, with some promising results published this year. “I tend to think of the field as primarily discovery-driven, and the biggest breakthrough may and likely will come unexpectedly, as we continue to pursue existing research directions,” he says. However, by exploring kagome lattice materials, it seems they are on the verge of another groundbreaking discovery.

Physicists hope that topological materials could eventually find applications in faster, more efficient computer chips or fanciful quantum computers. But the real reward of topological physics is a deeper understanding of the nature of matter itself. “I have long thought of ways to use topological materials to make analogs of black holes or wormholes in the lab but did not get a chance to dedicate to these ideas,” Hasan says. “Emergent phenomena in topological physics are probably all around us, even in a piece of rock.” He recalls a poet of William Blake which aptly describes his research endeavors:

To see a World in a Grain of Sand

And a Heaven in a Wild Flower

Hold Infinity in the palm of your hand

And Eternity in an hour

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