  The states of two electrons in tunnel-coupled semiconductor quantum dots can
be described in terms of a two-spin Hamiltonian with an isotropic Heisenberg
interaction. A similar description does not apply in the case of holes due to
their multiband character and spin-orbit coupling, which introduces a strong
anisotropy in the angular-momentum space and mixes orbital and spin degrees of
freedom. Here we investigate two-hole states in coupled Si and Ge quantum dots
within a 6-band $\boldsymbol{k}\cdot\boldsymbol{p}$ and
Configuration-Interaction approach. We integrate and interpret the numerical
results by means of an effective 4-band Hubbard model, which provides general
expressions for the two-hole eigenstates. The spin-orbit mixing in the single-
and two-hole states is quantified via the computation of the linear entropy of
the reduced spin density operator. We find that the ground state and first
excited multiplet of the two-hole system are, respectively, a singlet and a
triplet; however, differently from the one-band case, they are characterized by
a combination of states belonging to different $J$ subspaces, with dominant
contributions coming respectively from even and odd values of $J$. Finally, by
combining our numerical results with the effective model, we illustrate a
pseudospin-$1/2$ description of the single-hole states which accounts for the
main features of the lowest two-hole states in the regime of weak spin-orbit
entanglement.
