Researchers Solve First Problem From Mathematical Physics Wish List

Quantum conductance under extreme conditions

In the years following von Klitzing’s experiment, when researchers decided to study the Hall conductance under extreme conditions — those involving even lower temperatures and higher strength magnetic fields — “the Hall conductance was quantized in fractional multiples of what had been previously observed. It’s as if somehow electrons themselves were being split up into smaller particles, each carrying a fraction of the electron’s charge,” the news release notes.

As a result of that work, three researchers — Horst Störmer, Daniel Tsui and Robert Laughlin — shared the 1998 Nobel Prize in Physics. However, they still didn’t understand how the electrons acted together to create the results they observed. And that’s the focus of the problem Hastings and Michalakis recently solved.

The math of stretchy, bendy objects

They used topology, the mathematical study of the properties of objects that don’t change when the objects are bent or stretched, to reach their solution. One thing that isn’t allowed in topology? Tearing.

In the wacky world of topology, a donut can be stretched until it becomes a coffee cup. However, it can’t become, say, a sphere, because that would require tearing.

Other researchers had already explored this problem from a topological perspective, but their work made two assumptions that aren’t present in the new solution.

Electrons lose their identify in a topological state of matter, making them spread out more, and creating a “stable, entangled system that acts like a single object,” Michalakis noted in the news release. Prior work assumed that this behavior, which explains the global properties of quantum Hall conductance, also explains the properties present at a local level. And the other assumption was that the electrons in the system weren’t interacting with each other.

Michalakis and Hastings found a way to remove both assumptions, by”connecting the global picture to the local picture” that clarifies how the quantum Hall system works.

“The Hall conductance, it turns out, is equal to the number of times that path winds around the topological features of the mathematical shape describing the quantum Hall system,” Michalakis noted. The researchers now understand why the Hall conductance is an integer multiple and why impurities don’t prevent the conductance from occurring.

Their research was published in the journal Communications in Mathematical Physics and the problem has been designated as “solved” on the wish list of mathematical physics problems. Future applications of the work could include better understanding of quantum computing and other areas of quantum science.

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