Topological phases and K-theory


Topological K-theory has shown up in the classification of symmetry-protected topological phases of free fermions. This started with the work of Kitaev, and was later continued by Freed and Moore, who modified K-theory to twisted K-theory in order to fit the physics. In more recent years, mathematically rigorous tools have been developed for the computation of this twisted K-theory. We present an introductory approach to both the mathematical and physical side of the subject, notably without assuming a prior knowledge of K-theory. The main results of Freed and Moore are summarised for the symmorphic case, and the basic cohomological tools for the computation of these K-groups are outlined. These tools are then applied to specific calculations. We classify all possible one-dimensional topological phases in all ten Altland–Zirnbauer classes, under a few assumptions. We do this mainly by using the Mayer–Vietoris exact sequence, and we find that for one-dimensional systems this method coincides with the Atiyah–Hirzebruch spectral sequence approach. We also treat the three-dimensional space group $F222$ in class A, i.e., topological phases protected by crystal symmetries only. This was already done by Gomi et al., but not in much detail; we provide a detailed version of this calculation. Using the Atiyah–Hirzebruch spectral sequence, we find the same result; in particular, we find a $\mathbf{Z}/2$-invariant.

This is my Bachelor’s thesis, which was supervised by Hessel Posthuma and Jan de Boer.

There are some errata for this thesis (mostly typos):

Sven van Nigtevecht
PhD student