**Ten kinds of matter**A cool discovery: substances can be divided into 10 kinds.

The basic idea is pretty simple. Some substances have

**time-reversal symmetry**: they would look the same, even on the atomic level, if you made a movie of them and ran it backwards. Some don't - these are more rare, like certain superconductors made of yttrium barium copper oxide! The substances that

*do* have time reversal symmetry have a symmetry operator T that can square to 1 or to -1: please take my word for this, it's a quantum thing. So, we get 3 choices, which are listed in the chart under T as 1, -1, or 0 (no time reversal symmetry).

Similarly, some substances have

**charge conjugation symmetry**, meaning a symmetry where we switch particles and holes: places where a particle is missing. The 'particles' here can be rather abstract things, like

**phonons** - little vibrations of sound in a substance, which act like particles - or

**spinons** - little wiggles in the spin of electrons. Basically any sort of wave can, thanks to quantum mechanics, also act like a particle. And sometimes we can switch particles and holes, and a substance will act the same way!

The substances that

*do* have charge conjugation symmetry have a symmetry operator C that can square to 1 or to -1. So again we get 3 choices, listed in the chart under C as 1, -1, or 0 (no charge conjugation symmetry).

So far we have 3 × 3 = 9 kinds of matter. What is the tenth kind?

Some kinds of matter don't have time reversal or charge conjugation symmetry, but they're symmetrical under the

*combination* of time reversal and charge conjugation! You switch particles and holes and run the movie backwards, and things look the same!

This chart shows a 1 under the S when your matter has this combined symmetry, and 0 when it doesn't. So,

**0 0 1** is the tenth kind of matter (the second row in the chart).

This stuff was first discovered around 1997 by Altland and Zirnbauer. But it's just the beginning of an amazing story. Since then people have found substances called

**topological insulators** that act like insulators in their interior but conduct electricity on their surface. We can make 3-dimensional topological insulators, but also 2-dimensional ones (that is, thin films) and even 1-dimensional ones (wires). And we can theorize about higher-dimensional ones, though this is mainly a mathematical game.

So we can ask which of the 10 kinds of substance can arise as topological insulators in various dimensions. And the answer is: in any particular dimension, only 5 kinds can show up. This chart shows how it works for dimensions 1 through 8. The kinds that

*can't* show up are labelled 0.

(There's more information in this chart, which I'm too lazy to explain now.)

If you look at the chart, you'll see it has some nice patterns. And

*it repeats after dimension 8*. In other words, dimension 9 works just like dimension 1, and so on.

There is a

*huge* amount of cool math lurking here, and you can see some more in my blog article:

http://golem.ph.utexas.edu/category/2014/07/the_tenfold_way.htmlThis math is called the

**ten-fold way**.

The chart here comes from the paper that showed only 5 kinds of topological insulator are possible in each dimension:

• Shinsei Ryu, Andreas P Schnyder, Akira Furusaki, and Andreas W. W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy,

*New J. Phys.* **12** (2010) 065010,

http://arxiv.org/abs/0912.2157.

#spnetwork arXiv:0912.2157

#must_read #condensed_matter #topology