Exact Rokhsar-Kivelson points hosting first-order phase transitions in $S=1$ compass models

We study $S=1$ compass models on the square and kagome lattices with bond-dependent bilinear and biquadratic interactions parametrized by an angle $θ$. For compatible geometries, we determine the exact ground-state manifolds throughout the interval $π/4\leqθ\leq3π/4$. At each endpoint, a product of on-site unitaries maps the Hamiltonian to the Laplacian of a configuration graph $\mathcal{G}$ in the $S=1$ Cartesian basis. These endpoints are exact Rokhsar-Kivelson (RK) points, with one ground state per dynamically connected component of $\mathcal{G}$. Each such state is an equal-amplitude superposition in the transformed basis and acquires configuration-dependent phases upon transforming back to the original basis. Both models possess $m=O(L)$ independent and mutually commuting conserved $\mathbb{Z}_2$ line charges, where $L$ is the linear system size, forcing a ground-state degeneracy of at least $2^m$ at the RK endpoints. Throughout the interior region, $π/4<θ<3π/4$, the ground-state manifold on each lattice is instead spanned by two product states, both annihilated by the Hamiltonian. On both lattices, each RK endpoint also hosts a first-order transition into this intervening interior phase spanned by the two product states with long-range order.