Classification of Bogoliubov de Gennes Hamiltonians

Question

This question refers to the paper Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures by Altland and Zirnbauer. In the paper the authors give a classification of Bogoliubov de Gennes Hamiltonians. More precisely the authors state that

The aim of the current section is to classify systems according to their symmetries. Using the BdG formalism we will show that the presence or absence of time-reversal and/or spin-rotation invariance leads to four distinct symmetry classes.

While I do understand their classifications I am wondering if it is sufficient to restrict oneself to time-reversal and spin-rotation invariance only? I am wondering why the authors seem to disregard important symmetries such as inversion symmetry of the underlying lattice or other point group symmetries? Hence I am wondering if the classification given in the paper is in fact complete? In particular the presence and absence of inversion symmetry plays a key role in the theoy of superconductivity (centrosymmetric vs. non-centrosymmetric superconductors)

I would be very happy to hear thoughts and opinions on that.

Answer 1

The reason that space group is not considered by Altland and Zirnbauer is that they were interested in the implication of the symmetries for transport properties of disordered electrons, and once you have disorder, spatial symmetries are no longer there. But you are right that the classification can be extended to include inversion symmetry and similarly for point group symmetries. For a modern view on the classification, see http://arxiv.org/abs/0912.2157. For inversion symmetry, one recent work is http://arxiv.org/abs/1403.5558. For other point group symmetry you can just search "topological crystalline insulators". A recent review is http://arxiv.org/abs/1501.00531.