# What are examples of condensed matter systems of all 10 classes listed in the 10-fold way?

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The tenfold way is a mathematical classification of Hamiltonians used in condensed matter physics, based on their symmetries.  While it has connections to many mathematical subjects, I'd like to know examples of real-world condensed matter systems of all ten kinds.  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 we can assume without loss of generality that $S^2 = 1$.

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

asked Jan 23, 2023 in Q&A

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