# 40 Years of Quantum Leap: An Important Turning Point in Condensed Matter Physics

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**The ****D****iscovery of ****T****he ****Q****uantum Hall ****E****ffect**

The story begins at the end of the 19th century.

In 1879, the physicist Edwin Hall discovered for the first time in a groundbreaking experiment that when a magnetic field acts vertically on a metal strip, it will cause electrons to gather at both ends of the metal strip, making the metal The current is deflected to form a measurable voltage. This phenomenon is called the Hall effect, and the resulting voltage perpendicular to the direction of the current is called the Hall voltage.

However, the classic Hall effect is not the subject of this article. What we are talking about today is the quantum version of the Hall effect that was accidentally discovered a century later-the quantum Hall effect.

In 1980, the German physicist Klaus von Klitzing was conducting an experiment. He exposed atomically thick crystalline materials to a strong magnetic field at low temperatures and found that as the strength of the magnetic field increases, the increase in metal conductance does not increase smoothly and gradually as predicted by classical physics, but is quantized. Ascend step by step. Von Klitzing realized that in this case, the Hall resistance value is related to two fundamental constants, one of which is Planck's constant h, and the other is the electronic charge e: the quantized Hall resistance value is proportional to integer multiples At h/e².

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**Topological ****C****oncept of ****Q****uantum Hall ****E****ffect**

The discovery of the quantum Hall effect occurred in the 1980s. They all originated from experiments, and then related theories were developed. The physics at that time could not fully explain why the resistance would undergo such discrete jump changes with the change of the magnetic field. Solis used the concept of topology to challenge the theory of the electrical conductivity of materials at that time and proposed a breakthrough new theory.

Topology describes that unless an object is torn, it will remain unchanged no matter how stretched, distorted or distorted. This is somewhat similar to the quantum Hall effect: even if there are impurities in the material, the conductance will not change. Therefore, these topological objects with the number of holes of one, two, three, four... are "borrowed" to describe the conduction phenomenon in the quantum Hall effect.

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**Topology ****B****ased on ****M****athematical ****P****roof**

However, mathematicians do not seem to be satisfied with the explanation of the quantum Hall effect made by Solis and others. For mathematicians, the mechanism behind it is still an unsolved mystery. In 2015, the mathematician Spirizon Mihalakis of the California Institute of Technology and the physicist Matthew Hastings of Microsoft published a rigorous mathematical proof. However, Mikhailakis and Hastings succeeded in establishing an indestructible connection between topology and quantum Hall effect. They connected the overall picture with the local picture in a novel way and successfully settled these assumptions.

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**The ****C****ombination of ****T****opological ****S****tructure ****A****nd ****Q****uantum Hall ****E****ffect**

However, the story is not over yet. Topology has been favored by more and more physicists. For example, physicists and topologists now cooperate to study the fractional quantum Hall effect.

In 1939, Paul Dirac, one of the founders of quantum physics, said in a lecture: "The connection between pure mathematics and physics is becoming closer." He even thought that these two disciplines might eventually The two merge into one, so that every branch of pure number has its physical application. This view naturally arouses the dissatisfaction of some pure mathematicians, because it sounds as if physicists are just viewing mathematics as a tool for them to study the natural world. Although Dirac’s words may be questionable, the combination of pure mathematics and physics can indeed bring unexpected gains. The mathematical proof of quantum Hall resistance is the best example.

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