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We propose two new strategies to construct a family of non-integrable spinchains with exactly solvable subspace based on the idea of quasiparticleexcitations from the matrix product vacuum state. The first one allows theboundary generalization, while the second one makes it possible to constructthe solvable subspace with interacting quasiparticles. Each generalization isrealized by removing the assumption made in the conventional method, which isthe frustration-free condition or the local orthogonality, respectively. Wefound that the structure of embedded equally-spaced energy spectrum is notviolated by the diagonal boundaries, as log as quasiparticles arenon-interacting in the invariant subspace. On the other hand, we show thatthere exists a one-parameter family of non-integrable Hamiltonians which showperfectly embedded energy spectrum of the integrable spin chain. Surprisingly,the embedded energy spectrum does change by varying the free parameter of theHamiltonian. The constructed eigenstates in the solvable subspace are thecandidates of quantum many-body scar states, as they show up in the middle ofthe energy spectrum and have entanglement entropies expected to obey thesub-volume law.

Exactly solvable subspaces of non-integrable spin chains with boundaries and quasiparticle interactions