## Can electron be split in topological insulators?

I was shocked when I saw John Preskill’s tweet shown above. I am no expert in condensed matter physics. I was a professional physicist in the past specializing in beam-beam interactions (accelerator physics) and I follow particle physics very closely. Preskill’s statement sounded outrageous. I was very curious. Then, I found an article written by Charles L. Kane who received the Breakthrough Prize in Fundamental Physics in 2019. In the article he explains that these “impossible” electrons are emergent particles in the same way “holes” are.

You can rest assured. The electron is indivisible and can’t be split.

This does not diminish the significance of topological insulators which started a revolution in condensed matter physics and material science.

Splitting the Indivisible

by Charles L. Kane (University of Pennsylvania) published at the website of Simons Center for Geometry and Physics (Stonybrook University)

“The simplest version of a topological insulating phase occurs in a one-dimensional (1D) polymer called polyacetylene, which is an electrical insulator that consists of a chain of atoms with alternating strong and weak bonds. However, there are two possible configurations (“strong-weak-strong-weak” and “weak-strong-weak-strong”) which are topologically different in the above sense. The cartoon picture for this insulator is that there are twice as many spaces for the electrons than there are electrons, so in the A-phase (B-phase) the electrons occupy only the black (red) sites in Fig. 1c. Importantly, there are just enough electrons to exactly compensate the positive charge of the nuclei, so that with half the spaces filled the system is overall electrically neutral.

Now, something magic happens when you add an extra electron, say, to the A-phase. In Fig. 1c there is one added electron, so the net charge is -e. In Fig. 1d, by simply shifting electrons over one space, that added charge -e splits in half! The only places where there is any net charge is in the regions where the electrons are bunched together.  Since there are two of them they each have charge precisely -e/2These -e/2 charges can move and are emergent particles in the same way that holes are

How can this be? The electron is indivisible and can’t be split.  But notice that between the -e/2 charges the insulator is in the B phase. The “impossible” -e/2 charges exist on the boundary between topologically distinct insulators. This is the essence of topological phases of matter. They allow the indivisible to be split by putting the “impossible” on the boundary. There are many examples of this general phenomenon, and the impossible” things that they allow are truly remarkable.”

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