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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.
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