One of the world’s greatest unsolved mathematical problems has finally been beaten. Spiros Michalakis, a quantum physicist from Caltech’s Institute for Quantum Information and Matter (IQIM) and Matthew Hastings, a researcher at Microsoft, have now been credited with definitively solving a mathematical problem related to the “quantum Hall effect.”— the tendency for the electrical conductivity of materials to take on integer values at very low temperatures.

The mathematical problem is one of 13 unsolved physics problems first proposed by Princeton physicist Michael Aizenman in 1999. To date, this is the first of those 13 problems to be marked as definitively solved, with one other being marked as partially solved.

Although originally published in 2015, the dense 40-page long mathematical proof presented by Michalakis and Hastings took some time to be digested by the mathematical community. In their April 2018 newsletter, the International Association for Mathematical Physics (IAMP) announced the problem officially solved. The solution is important as it dramatically showcases the power and elegance of mathematical theorizing in the realm of physics. Physics is notable for being an extremely mathematical and abstract discipline and a number of advances in physics have come from the realm of pure mathematics. Michalakis hopes that the discovery will “invigorate interest in the field of mathematical physics.”

## Quantum Hall Effect: Strange Electron Behavior

The original Hall effect was first described in 1879 by American physicist Edwin Hall. Hall noticed that a voltage was produced across an electrical conductor when he introduced a magnetic field with a vector perpendicular to that of the conductor current. The explanation for this effect is that the magnetic field deflects the straight path of electrons, causing them to build up disproportionately on one side of the conductive surface, producing a voltage across the conductor transverse to the flow of current. The discovery of the original Hall effect was notable as it offered the first real proof that electrical currents were constituted out of the movement of electrons, not protons.

In the 1980s, experiments by German physicist Klaus Von Klitzing discovered something interesting. Hall’s original equations describing the effect predicted that the electrical conductance of the Hall material would increase linearly and proportionally to the strength of the magnetic field. Von Klitzing discovered, however, that at extremely low temperatures electrical conductance did *not* increase linearly with respect to the magnitude of the magnetic field. Instead, electrical conductance increased in a step-by-step fashion, taking on different integer values. Von Klitzing observed that under a particular thermodynamic limit, the electrical conductance of the Hall conductor would take on discrete quantized values that held regardless the electron density of the current. This finding is strange for, as Hasting notes, “These impurities are randomly distributed in the material so you might think they would have a random effect on the conductance, but they don’t.”

To put the problem in more general terms, classical physics predicts that electrical conductance should increase or decrease continuously. Von Litzing discovered that, on the contrary, electrical conductance would take on stable discrete values regardless of the density of electrons in the current. The problem is striking to physicists as it seems on the one hand, at certain temperatures a group of electrons act like a single entity and exhibit one global property. On the other hand, it is common sense and accepted theory that electrical currents are composed out of multiple independent entities each with their own unique dynamics and properties. The main mathematical problem solved by the duo was how to bridge the gap between these views and explain how a fluctuating quantity of electrons could exhibit the stable and robust conductance values that Von Klitzing observed.

## The Solution

The key to the solution was found in topology, a field devoted to describing the mathematical properties of geometric surfaces. One of the things that topology studies is what are known as “topological invariants”—properties of geometric surfaces that remain invariant under certain changes to those surfaces. The key insight to the proof was that something similar is going on in the quantum Hall effect; the electrical conductance of the material is invariant with respect to changes in the electron density across that material.

A common maxim in electrical engineering is that “electricity follows the path of least resistance,” Electrical currents by nature will seek out the paths that are the easiest to follow. Essentially, Michalakis and Hastings proved that quantum Hall systems have a characteristic “path” that, under a certain scale (i.e. quantum scale) electrons will always follow, regardless of the number of electrons around.

To illustrate this point, imagine you are observing the earth from space. At that level of resolution, it appears that the globe is smooth and that one could travel around it relatively unimpeded. Of course, when one gets closer they see obstacles like mountains and valleys that would impede their path. What Michalakis and Hastings mathematically proved was, that for any possible curved geometric surface, there exists at least one flat “path” that will not encounter any dips or peaks. It just so happens that the number of times that path circles around the surface of a quantum Hall system is exactly equal to the Hall conductance for that surface. In other words, the surface of each Hall conductor has a privileged “path” that, under a certain thermodynamic limit, all electrons will follow, thus manifesting as stable integer conductance values. According to Hastings, “Impurities are like small detours you decide to take from the ‘golden’ path, as you travel around the world. They won’t affect how many times you decide to go around the globe.”

The particular proof is of importance as it once again demonstrates the scientific fecundity of abstract mathematical techniques. Physics and mathematics have long shared an intimate relationship, one that only seems to deepen with every advancement in either field. In fact, this seems to be precisely the sentiment of Michalakis who proclaimed, “as is often the case with proofs of significant problems in math, the solution leads to new ideas and techniques that open the doors to resolving several other important questions.” Michalakis also maintains that the new understanding could open up possible avenues of research and applications in quantum computing and other quantum technologies like superconductors.

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