Tuesday, August 1, 2017

The role of the Platonic ideal in solid state physics

In the book Who Got Einstein's Office?, about the Institute for Advanced Study at Princeton, the author Ed Regis, mocks it as the "One True Platonic Heaven" because he claims its members are Platonic idealists, who are interested in pure theory, and disdain such "impurities" as computers and applied mathematics.

This stimulated me to think about the limited but useful role of pure mathematics, Platonic idealism, and aesthetics in solid state theory. People seem particularly excited when topology and/or geometry plays a role.

The first example I could think of is the notion of a perfect crystal.

Then comes Bloch's theorem, which surely is the central idea of introductory solid state physics.

Beautiful examples where advanced pure maths plays are role are
Chern-Simons theory of edge states in the Quantum Hall Effect
and topological terms in the action for quantum spin chains, as elucidated by Haldane.

As I have said before I think topological insulators is a beautiful, fascinating, and important topic. However, I am concerned by the disproportionately large number of people working on the topic and the associated hype. I wonder if some of the appeal and infatuation is driven by Platonic idealism.

For a classic example of how Platonism leads to imperfect theory is Kepler's Platonic solid model of the Solar System from Mysterium Cosmographicum (1596).

Good theory finds a balance between beauty and the necessity of dirty details.

Can you think of other examples where Platonic idealism plays a positive role in condensed matter theory?


  1. Could you explain why you say that the field of topological materials has evolved to have some "Platonic Idealism" in your opinion? I study them as a graduate student so they are essentially what I spent most my time on.

    1. My subjective judgement is that some of the appeal, particularly to theorists, is that there is some beautiful mathematics involved: e.g. Chern numbers and topological invariants.

    2. For more on this see the Nature article

      It ends with

      "It is this interplay between maths and physics that lies at the heart of the field, says Kane: “What drives me is the intersection of something which is both incredibly beautiful, and also comes to life in the real world.”"

  2. The Gaussian model of polymers, a source of rich mathematics in its random walk incarnation, but also a perfect starting point to think about the physical aspects of polymers.