For classifying Hamiltonians $H(\vec{k})$ of topological insulators/superconductors in the tenfold way, one has to see whether the Hamiltonians obeys (disobeys) symmetries of the following type (let's ignore chiral symmetry and only focus on either Time-reversal (TRS) or Particle-hole (PHS)
$UH(\vec{k})^*/U = \pm H(-\vec{k})$,
where $U$ is unitary. I understand, that in a $d$-dimensional system ($\vec{p}\in \mathbb{R}^d$) these symmetries are straightforward. However, there are ways to introduce variables, which act similarily to momenta but aren't physical momenta, like the superconducting phases $\vec{\varphi}$ in a multiterminal Josephson Junction. It has been shown in this work that a system containing 4 superconductors can have a non-trivial Chern number given by the same formula as for topological insulators in $d=2$.
However, as stated in the above paper, the tenfold way can't be used to classify such a system because of the "unconventional" PHS in multiterminal Josephson junctions, namely
$UH(\vec{\varphi})^*/U = - H(\varphi)$,
where there is no "-"-sign for the argument of the Hamiltonian in the right side of the equation in constrast than for a band insulator. I wanted to see where exactly this circumstance results in a concrete difference and tried to follow each step in Kitaev's work. However, I am not sure where exactly I obtain a problem as it is not clear to me where exactly the fact that $\vec{k}\to -\vec{k}$ appears in the classification procedure.
I would be appreciate some insight, where one example of the tenfold way and its homotopy group is rigorously derived with every single step.
