into the 10 Altland-Zirnbauer (AZ) symmetry classes [@Altland1997Nonstandard; @heinzner2005symmetry; @zirnbauer2010symmetry] and form the “tenfold way". Each AZ symmetry class corresponds to a topological “classifying space" of possible free-fermion Hamiltonians respecting the symmetry, and these are the topological invariants of these classifying spaces which are the ${\mathbb{Z}}$ and ${\mathbb{Z}}_2$ topological indices.

There are many different view-points for this profound classification [@Schnyder2008Classification; @schnyder2009classification; @ryu2010topological; @kitaev2009periodic; @stone2010symmetries]. The approach of Refs. \[\], which was pioneered by Kitaev \[\], takes an algebraic perspective. In this paradigm, the set of non-spatial symmetry transformations forms a Clifford algebra with a specific action over the Hilbert space. The enumeration of all possible topologically distinct Hamiltonians is equivalent to asking the following: How many actions can a Hamiltonian have on the Hilbert space which are compatible with the algebra of non-spatial symmetries? The answer to this question is exactly the “classifying space" for the AZ symmetry class; see Fig. \[fig:diag1\]. The twofold and eightfold periodic structures within the table then naturally follow from the Bott periodicity of Clifford modules [@atiyah1964clifford].

[ccccccc]{} &

  -------------
   Non-spatial
   symmetries
  -------------

& &

  -------------
   Symmetries
        +
   Hamiltonian
  -------------

& & &

  -------------
   Classifying
      space
  -------------

\
& $\Downarrow$ & & $\Downarrow$ & & 