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Is there a reference that systematically derives the topological invariant/winding number for all the ten symmetry classes in Altland and Zirnbauer's periodic table? For example, in this answer, there is the mention of the standard Chern number for the D class, but a slightly different winding number for the AIII class.

In general, given a gapped Bloch Hamiltonian that respects/breaks certain symmetries, how does one go about finding this winding number? Any insight would be helpful, even in specific cases. Thanks!

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There is a beautiful paper by Grossmann and Schulz-Baldes that derives this: https://link.springer.com/article/10.1007/s00220-015-2530-6?wt_mc=internal.event.1.SEM.ArticleAuthorOnlineFirst

In principle for class A (no symmetry) there are standard formula for the Chern number in all even dimensions which are generalizations of the formula of the integral of the Berry curvature.

In class AIII (chiral) there is an analog of that for unitary maps.

The other rows of the table are much more complicated.

The main procedure to calculate the invariant (of a bulk system) passes through the main stage of solving the system, which is, to calculate its Fermi projection. This is a highly non-trivial step that is essentially equivalent to diagonalization the Hamiltonian. Once this is accomplished, there are standard formulas in which you can plug in that projection and get a number, and they are tabulated in the paper above.

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