This appendix records the minimal $D_{2h}$ point-group bookkeeping used later when discussing orthorhombic multiorbital Hamiltonians. It is a local point-group guide, not a replacement for a full space-group or little-group analysis of a nonsymmorphic material [1, 2].

The conventional assignments for the coordinate axes used in this thesis are

Thus polar-vector components transform as

while axial-vector components transform as

Spin components transform in the same way as angular momentum,

Because $D_{2h}$ is Abelian, all irreducible representations are one-dimensional. Direct products can therefore be computed by multiplying the inversion parity and the three $B_i$ labels. Useful examples are

and

For an onsite interorbital SOC term of the form

the invariant condition is

Therefore an onsite ${\tau }_y{\sigma }_x$ term is allowed only if

Equivalently, if the two retained orbitals transform as ${\Gamma }_a$ and ${\Gamma }_b$, then the corresponding interorbital matrix element is allowed when

For a momentum-dependent term,

the condition becomes

For example, a $k_y{\tau }_y{\sigma }_x$ term has $\Gamma [k_y]=B_{2u}$ and is allowed if

Since $B_{2u}\otimes B_{3g}=B_{1u}$, this requires

The same logic applies to all Pauli-matrix terms in an effective Hamiltonian. The caveat is that nonsymmorphic space groups, including $Cmcm$, can enforce additional momentum-dependent degeneracies and compatibility relations. Those require the full little-group representation at the relevant $\mathbf{k}$, not only the local $D_{2h}$ multiplication table.