Welcome to the Weird Mathematical World of Topology

But physicists considered the quantum Hall effect to be a peculiar situation, a one-off, says Allan MacDonald, a physicist at the University of Texas at Austin who studies the effect. Kane’s doughnut discovery extended Thouless’ work dramatically. “He showed that topology could apply not just in the very special circumstance, but very generally,” says MacDonald. That allowed Hasan and his team to make the first 3D topological insulator.

This has tangible benefits as well. Topological insulators have another unusual property. Guided by topology, which downplays details, electrons flowing through these materials don’t mind bumping into imperfections or defects. They don’t tend to lose energy and give off heat, as an electron flowing through a wire would. That means electronics made of these materials could, in theory, consume less power and become much more efficient.

And that’s just one of the technologies that topological materials could energize.

Mathematica ex Machina

If Chetan Nayak ever builds the computer of his dreams — one inspired by topology — he could become the world’s most dangerous hacker, stealing credit card numbers with ease. Or, if he used his powers for good, he could create a search engine light-years ahead of Google, help chemists design new drugs and aid physicists in understanding the building blocks of the universe.

These are some of the long-standing promises of quantum computing, which seeks to store bits of information not as 1s and 0s, as is the case in conventional computers, but in weird quantum states that can be partly 1 and partly 0 at the same time.

But you don’t need to worry about Nayak; he’s a researcher, not a hacker. And despite decades of work, quantum computers have yet to live up to their potential. Google has created a chip that has 72 quantum bits; IBM’s best effort sports 50; and Intel has a 49-qubit device. None of these machines can do anything more than the $200 chip in your laptop, packed with billions of transistors. Quantum devices are still puny, and significant barriers remain to supersizing them.

The problem is that these futuristic devices store information in fragile quantum states of individual subatomic particles. And these states are notoriously fickle; the slightest disturbance or defect can easily corrupt their information. This limits the computing power of quantum devices, which must spend most of their resources on correcting errors caused by contact with the outside world.

Nayak is betting topology can solve this problem by protecting quantum information from the outside world. “On paper, topology can be exploited,” he says. His employer, Microsoft, agrees and has set up a facility called Station Q at the University of California, Santa Barbara, dedicated to building a new kind of quantum computer. Step inside, and you’ll find chalkboards covered in the equations of topology, written by physicists and mathematicians just steps away from the Pacific Ocean.

Despite ramping up its efforts in the past year, Station Q has yet to make a chip that draws on the special benefits of topology. “The experiments aren’t quite there,” Nayak says. But he’s confident in the approach.

Miniature Universes

Thanks to the unusual mathematics that governs their behavior, topological materials effectively house miniature universes that play by rules different from the outside world. Particles that do not exist in nature can appear in strange forms inside these materials.

Consider the Weyl fermion. Nearly 90 years ago, while playing around with the equations of quantum physics, as one does, German physicist Hermann Weyl showed that this massless and charged particle could, in theory, exist. But it has never shown itself among the elementary particles that make up the universe or appeared in experiments searching for new particles by smashing other particles together.