Exactly solvable subspaces of nonintegrable spin chains with boundaries and quasiparticle interactions

We propose two strategies to construct a family of nonintegrable spin chains with exactly solvable subspaces based on the idea of quasiparticle excitations from the matrix product vacuum state [Haegeman et al., Phys. Rev. B 88, 075133 (2013)]. 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 [Moudgalya et al., Phys. Rev. B 102, 085120 (2020)], which is the frustration-free condition or the local orthogonality, respectively. We found that the structure of the embedded equally spaced energy spectrum is not violated by the diagonal boundaries, as long as quasiparticles are identical and noninteracting in the invariant subspace. On the other hand, we show that there exists a one-parameter family of nonintegrable Hamiltonians, which shows the perfectly embedded energy spectrum of the integrable spin chain. Surprisingly, the embedded energy spectrum does not change by varying the free parameter of the Hamiltonian. The constructed models weakly break ergodicity, in which strong ETH is expected to be violated.