Integrability and non-invertible symmetries of projector spin chains

We show that nearest-neighbor spin chains composed of projectors to 2-qudit product states are integrable. The $R$-matrices (with a multidimensional spectral parameter) include additive as well as non-additive parameters. They satisfy the colored Yang-Baxter equation. The local terms of the resulting Hamiltonians exhaust projectors with all possible ranks for a 2-qudit space. The Hamiltonian can be Hermitian or not, with or without frustration. The ground state structures of the frustration-free qubit spin chains are analysed. These systems have either global or local non-invertible symmetries. In particular, the rank 1 case has two product ground states that break global non-invertible symmetries (analogous to the case of the two ferromagnetic states breaking the global $\mathbb{Z}_2$ symmetry of the $XXX$ spin chain). The Bravyi-Gosset conditions for spectral gaps show that these systems are gapped. The associated Yang-Baxter algebra and the spectrum of the transfer matrix are also studied.