Supercell symmetry modified spectral statistics of Kramers-Weyl fermions

We calculate the spectral statistics of the Kramers-Weyl Hamiltonian $H=v\sum_{\alpha} \sigma_\alpha\sin p_\alpha+t \sigma_0\sum_\alpha\cos p_\alpha$ in a chaotic quantum dot. The Hamiltonian has symplectic time-reversal symmetry ($H$ is invariant when spin ${\sigma }_{\alpha }$ and momentum $p_{\alpha }$ both change sign), and yet for small $t$ the level spacing distribution $P(s)\propto s^{\beta }$ follows the $\beta =1$ orthogonal ensemble instead of the $\beta =4$ symplectic ensemble. We identify a supercell symmetry of $H$ that explains this finding. The supercell symmetry is broken by the spin-independent hopping energy $\propto t\cos p$, which induces a transition from $\beta =1$ to $\beta =4$ statistics that shows up in the conductance as a transition from weak localization to weak antilocalization.