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

We propose two new strategies to construct a family of non-integrable spin chains with exactly solvable subspace based on the idea of quasiparticle excitations from the matrix product vacuum state. The first one allows the boundary generalization, while the second one makes it possible to construct the solvable subspace with interacting quasiparticles. Each generalization is realized by removing the assumption made in the conventional method, which is the frustration-free condition or the local orthogonality, respectively. We found that the structure of embedded equally-spaced energy spectrum is not violated by the diagonal boundaries, as log as quasiparticles are non-interacting in the invariant subspace. On the other hand, we show that there exists a one-parameter family of non-integrable Hamiltonians which show perfectly embedded energy spectrum of the integrable spin chain. Surprisingly, the embedded energy spectrum does change by varying the free parameter of the Hamiltonian. The constructed eigenstates in the solvable subspace are the candidates of quantum many-body scar states, as they show up in the middle of the energy spectrum and have entanglement entropies expected to obey the sub-volume law.