### Question on Condensed Matter Physics

#### Posted by John Baez

The tenfold way is a mathematical classification of Hamiltonians used in condensed matter physics, based on their symmetries. Nine kinds are characterized by choosing one of these 3 options:

- antiunitary time-reversal symmetry with $T^2 = 1$, with $T^2 = -1$, or no such symmetry.

and one of these 3 options:

- antiunitary charge conjugation symmetry with $C^2 = 1$, with $C^2 = -1$, or no such symmetry.

(Charge conjugation symmetry in condensed matter physics is usually a symmetry between particles - e.g. electrons or quasiparticles of some sort - and holes.)

The tenth kind has unitary “$S$” symmetry, a symmetry that simultaneously reverses the direction of time and interchanges particles and holes. Since it is unitary and we’re free to multiply it by a phase, we can assume without loss of generality that $S^2 = 1$.

**What are examples of real-world condensed matter systems of all ten kinds?**

## p-n junctions

I think p-n junctions would be the tenth type. Not confident though. https://en.wikipedia.org/wiki/P%E2%80%93n_junction