Fractional quantum Hall states in atom arrays
Our second approach to create a topological order in atom arrays is to focus on a different kind of topological order: fractional quantum Hall (FQH) states. These were first discovered in condensed matter. It is possible to confine electrons to move in two-dimensions only (such as in the 2D material graphene or in so-called metal-oxide-semiconductor transistors) and then put them in a strong perpendicular magnetic fields. The electrons then move in circles (so-called “cyclotron motion”), but since they are quantum objects, only some values of radius are allowed. Thus, the energy can only take certain fixed values (we call them “Landau levels”). There are however different possibilities of an electron having the same energy, because the center of the orbit can be located in different places – we say that Landau levels are “degenerate”. And when there is degeneracy, the interaction between electrons becomes very important. Without interactions, there are many possible ways of arranging electrons within a Landau level, all with the same energy. In the presence of interactions, some arrangements become preferred – and it turns out those correspond to topological orders known as the FQH states. Such systems host anyons which look like fractions of an electron – like somehow the electron split into several parts.
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